View source for Pythagorean theorem

{{short description|Equation relating the side lengths of a right triangle}}
{{about|classical geometry|the baseball term|Pythagorean expectation}}
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[[File:Pythagorean.svg|thumb|260px|'''Pythagorean theorem'''<br/>The sum of the areas of the two squares on the legs (''a'' and ''b'') equals the area of the square on the hypotenuse (''c'').]]
{{General geometry |concepts}}

In [[mathematics]], the '''Pythagorean theorem''', also known as '''Pythagoras' theorem''', is a fundamental relation in [[Euclidean geometry]] among the three sides of a [[right triangle]]. It states that the area of the square whose side is the [[hypotenuse]] (the side opposite the [[right angle]]) is equal to the sum of the areas of the squares on the [[Cathetus|other two sides]]. This [[theorem]] can be written as an [[equation]] relating the lengths of the sides ''a'', ''b'' and ''c'', often called the "Pythagorean equation":<ref name="Sally0">{{cite book |title=Roots to research: a vertical development of mathematical problems |author1=Judith D. Sally |author2=Paul Sally |page=63 |chapter=Chapter 3: Pythagorean triples |url=https://books.google.com/books?id=nHxBw-WlECUC&pg=PA63 |isbn=0-8218-4403-2 |year=2007 |publisher=American Mathematical Society Bookstore}}</ref>
:<math>a^2 + b^2 = c^2 ,</math>
where ''c'' represents the length of the hypotenuse and ''a'' and ''b'' the lengths of the triangle's other two sides.  The  theorem, whose history is the subject of much debate, is named for the [[Ancient Greece|ancient Greek]] thinker [[Pythagoras]].

The theorem has been given numerous proofs{{spndash}}possibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but ''n''-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons abound.

==Rearrangement proof==
[[File:Pythagoras-proof-anim.svg|thumb|The rearrangement proof (click to view animation)]]The two large squares shown in the figure each contain four identical triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the white space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem, [[Q.E.D.]]<ref>Benson, Donald. ''[https://books.google.com/books?id=8_vbuzxrpfIC&pg=PA172 The Moment of Proof : Mathematical Epiphanies]'', pp. 172–173 (Oxford University Press, 1999).</ref>

Heath gives this proof in his commentary on Proposition I.47 in Euclid's ''Elements'', and mentions the proposals of Bretschneider and Hankel that Pythagoras may have known this proof.  Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him."<ref>{{harvtxt|Euclid|1956}}, pp. 351–352</ref>  Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as a creator of mathematics, although debate about this continues.<ref>{{cite encyclopedia | last=Huffman | first=Carl | title=Pythagoras | encyclopedia=The Stanford Encyclopedia of Philosophy (Winter 2018 Edition) | editor-last=Zalta | editor-first=Edward N. | editor-link=Edward N. Zalta | url=https://plato.stanford.edu/archives/win2018/entries/pythagoras/}}, "It should now be clear that decisions about sources are crucial in addressing the question of whether Pythagoras was a mathematician and scientist. The view of Pythagoras’ cosmos sketched in the first five paragraphs of this section, according to which he was neither a mathematician nor a scientist, remains the consensus."</ref>

==Other forms of the theorem==
If ''c'' denotes the [[length]] of the hypotenuse and ''a'' and ''b'' denote the lengths of the other two sides, the Pythagorean theorem can be expressed as the Pythagorean equation:
: <math>a^2 + b^2 = c^2 .</math>
If the lengths of both ''a'' and ''b'' are known, then ''c'' can be calculated as
: <math> c = \sqrt{a^2 + b^2}. </math>
If the length of the hypotenuse ''c'' and of one side (''a'' or ''b'') are known, then the length of the other side can be calculated as
: <math>a = \sqrt{c^2 - b^2} </math>
or
: <math>b = \sqrt{c^2 - a^2}. </math>
The Pythagorean equation relates the sides of a right triangle in a simple way, so that if the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides, but less than their sum.

A generalization of this theorem is the [[law of cosines]], which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation.

==Other proofs of the theorem==
This theorem may have more known proofs than any other (the law of [[quadratic reciprocity]] being another contender for that distinction); the book ''The Pythagorean Proposition'' contains 370 proofs.<ref>{{Harv|Loomis|1968}}</ref>

===Proof using similar triangles===
[[File:Pythagoras similar triangles simplified.svg|thumb|Proof using similar triangles]]
This proof is based on the [[proportionality (mathematics)|proportionality]] of the sides of two [[Similarity (geometry)|similar]] triangles, that is, upon the fact that the [[ratio]] of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.

Let ''ABC'' represent a right triangle, with the right angle located at ''C'', as shown on the figure. Draw the [[altitude (triangle)|altitude]] from point ''C'', and call ''H'' its intersection with the side ''AB''. Point ''H'' divides the length of the hypotenuse ''c'' into parts ''d'' and ''e''. The new triangle ''ACH'' is [[Similarity (geometry)|similar]] to triangle ''ABC'', because they both have a right angle (by definition of the altitude), and they share the angle at ''A'', meaning that the third angle will be the same in both triangles as well, marked as ''θ'' in the figure. By a similar reasoning, the triangle ''CBH'' is also similar to ''ABC''. The proof of similarity of the triangles requires the [[triangle postulate]]: the sum of the angles in a triangle is two right angles, and is equivalent to the [[parallel postulate]]. Similarity of the triangles leads to the equality of ratios of corresponding sides:
:<math> \frac{BC}{AB}=\frac{BH}{BC} \text{ and } \frac{AC}{AB}=\frac{AH}{AC}.</math>
The first result equates the [[cosine]]s of the angles ''θ'', whereas the second result equates their [[sine]]s.

These ratios can be written as
:<math>BC^2 = AB \times BH \text{ and } AC^2=AB \times AH. </math>
Summing these two equalities results in
:<math>BC^2+AC^2=AB\times BH+AB\times AH=AB\times(AH+BH)=AB^2 ,</math>
which, after simplification, expresses the Pythagorean theorem:
:<math>BC^2+AC^2=AB^2 \ .</math>

The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another. One conjecture is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the ''Elements'', and that the theory of proportions needed further development at that time.<ref>{{Harv|Maor|2007| p= [https://books.google.com/books?id=Z5VoBGy3AoAC&pg=PA39&dq=%22why+did+Euclid+choose+this+particular+proof%22&hl=en&ei=WckoTLv4JIKknQecwvWoAQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCUQ6AEwAA#v=onepage&q=%22why%20did%20Euclid%20choose%20this%20particular%20proof%22&f=false  39]}}</ref><ref name=Hawking>
{{cite book |title=God created the integers: the mathematical breakthroughs that changed history |author=Stephen W. Hawking |page=12 |url=https://books.google.com/?id=3zdFSOS3f4AC&pg=PA12 |isbn=0-7624-1922-9 |year=2005 |publisher=Running Press Book Publishers |location=Philadelphia}}

This proof first appeared after a computer program was set to check Euclidean proofs.</ref>

===Euclid's proof===
[[File:Illustration to Euclid's proof of the Pythagorean theorem.svg|thumb|Proof in Euclid's ''Elements'']]
In outline, here is how the proof in [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' proceeds. The large square is divided into a left and right rectangle. A triangle is constructed that has half the area of the left rectangle. Then another triangle is constructed that has half the area of the square on the left-most side. These two triangles are shown to be [[Congruence (geometry)|congruent]], proving this square has the same area as the left rectangle. This argument is followed by a similar version for the right rectangle and the remaining square. Putting the two rectangles together to reform the square on the hypotenuse, its area is the same as the sum of the area of the other two squares. The details follow.

Let ''A'', ''B'', ''C'' be the [[Vertex (geometry)|vertices]] of a right triangle, with a right angle at ''A''. Drop a perpendicular from ''A'' to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs.

For the formal proof, we require four elementary [[lemma (mathematics)|lemmata]]:
# If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent ([[side angle side|side-angle-side]]).
# The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.
# The area of a rectangle is equal to the product of two adjacent sides.
# The area of a square is equal to the product of two of its sides (follows from 3).

Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square.<ref>See for example [http://www.slu.edu/classes/maymk/GeoGebra/Pythagoras.html Pythagorean theorem by shear mapping], Saint Louis University website Java applet</ref>
{{Clear}}
[[File:Illustration to Euclid's proof of the Pythagorean theorem2.svg|thumb|Illustration including the new lines]]
[[File:Illustration to Euclid's proof of the Pythagorean theorem3.svg|thumb|Showing the two congruent triangles of half the area of rectangle BDLK and square BAGF]]
The proof is as follows:
#Let ACB be a right-angled triangle with right angle CAB.
#On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in that order. The construction of squares requires the immediately preceding theorems in Euclid, and depends upon the parallel postulate.<ref name=Gullberg>

{{cite book |title=Mathematics: from the birth of numbers |author=Jan Gullberg |page=435 |url=https://books.google.com/books?id=E09fBi9StpQC&pg=PA435 |isbn=0-393-04002-X |year=1997 |publisher=W. W. Norton & Company}}

</ref>
#From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE at K and L, respectively.
#Join CF and AD, to form the triangles BCF and BDA.
#Angles CAB and BAG are both right angles; therefore C, A, and G are [[Line (geometry)#Collinear points|collinear]]. Similarly for B, A, and H.
#Angles CBD and FBA are both right angles; therefore angle ABD equals angle FBC, since both are the sum of a right angle and angle ABC.
#Since AB is equal to FB and BD is equal to BC, triangle ABD must be congruent to triangle FBC.
#Since A-K-L is a straight line, parallel to BD, then rectangle BDLK has twice the area of triangle ABD because they share the base BD and have the same altitude BK, i.e., a line normal to their common base, connecting the parallel lines BD and AL. (lemma 2)
#Since C is collinear with A and G, square BAGF must be twice in area to triangle FBC.
#Therefore, rectangle BDLK must have the same area as square BAGF = AB<sup>2</sup>.
#Similarly, it can be shown that rectangle CKLE must have the same area as square ACIH = AC<sup>2</sup>.
#Adding these two results, AB<sup>2</sup> + AC<sup>2</sup> = BD × BK + KL × KC
#Since BD = KL, BD × BK + KL × KC = BD(BK + KC) = BD × BC
#Therefore, AB<sup>2</sup> + AC<sup>2</sup> = BC<sup>2</sup>, since CBDE is a square.

This proof, which appears in Euclid's ''Elements'' as that of Proposition&nbsp;47 in Book&nbsp;1,<ref>[http://www.perseus.tufts.edu/cgi-bin/ptext?doc=Perseus:text:1999.01.0085:book=1:proposition=47 Elements 1.47] by Euclid. Retrieved 19 December 2006.</ref> demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares.<ref>

[http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI47.html Euclid's Elements, Book&nbsp;I, Proposition&nbsp;47]: web page version using Java applets from [http://aleph0.clarku.edu/~djoyce/java/elements/elements.html Euclid's Elements] by Prof. David E. Joyce, Clark University

</ref> This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used.<ref name=Hawking/><ref name=Pythagoras>The proof by Pythagoras probably was not a general one, as the theory of proportions was developed only two centuries after Pythagoras; see {{Harv |Maor|2007 |p= [https://books.google.com/books?id=Z5VoBGy3AoAC&pg=PA25&hl=en#v=onepage&q&f=false  25]}}</ref>

===Proofs by dissection and rearrangement===
We have already discussed the Pythagorean proof, which was a proof by rearrangement. The same idea is conveyed by the leftmost animation below, which consists of a large square, side {{nowrap|''a'' + ''b''}}, containing four identical right triangles. The triangles are shown in two arrangements, the first of which leaves two squares ''a''<sup>2</sup> and ''b''<sup>2</sup> uncovered, the second of which leaves square ''c''<sup>2</sup> uncovered. The area encompassed by the outer square never changes, and the area of the four triangles is the same at the beginning and the end, so the black square areas must be equal, therefore {{nowrap|''a''<sup>2</sup> + ''b''<sup>2</sup> {{=}} ''c''<sup>2</sup>.}}

A second proof by rearrangement is given by the middle animation. A large square is formed with area ''c''<sup>2</sup>, from four identical right triangles with sides ''a'', ''b'' and ''c'', fitted around a small central square. Then two rectangles are formed with sides ''a'' and ''b'' by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas ''a''<sup>2</sup> and ''b''<sup>2</sup>, which must have the same area as the initial large square.<ref>{{cite web |url=http://www.cut-the-knot.org/pythagoras/index.shtml#10 |title= Pythagorean theorem, proof number 10|author=[[Alexander Bogomolny]] |date= |work=Cut the Knot |publisher= |accessdate=27 February 2010}}</ref>

The third, rightmost image also gives a proof. The upper two squares are divided as shown by the blue and green shading, into pieces that when rearranged can be made to fit in the lower square on the hypotenuse – or conversely the large square can be divided as shown into pieces that fill the other two. This way of cutting one figure into pieces and rearranging them to get another figure is called [[dissection problem|dissection]]. This shows the area of the large square equals that of the two smaller ones.<ref name=specifics>{{Harv|Loomis|1968|loc= Geometric proof 22 and Figure 123| page= 113}}</ref>

{|
| [[File:Pythag anim.gif|thumb|Animation showing proof by rearrangement of four identical right triangles]]
| [[File:Pythagoras-2a.gif|thumb|Animation showing another proof by rearrangement]]
| [[File:Pythagorean theorem rearrangement.svg|thumb|Proof using an elaborate rearrangement]]
|}

===Einstein's proof by dissection without rearrangement===
[[File:Altitude of a right triangle.svg|thumb|Right triangle on the hypotenuse dissected into two similar right triangles on the legs, according to Einstein's proof]]
[[Albert Einstein]] gave a proof by dissection in which the pieces need not get moved.<ref>{{Cite book |last=Schroeder |first=Manfred Robert |title=Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise |publisher= Courier Corporation |year=2012|pages=3–4|isbn=0486134784|ref=harv}}</ref> Instead of using a square on the hypotenuse and two squares on the legs, one can use any other shape that includes the hypotenuse, and two [[similarity (geometry)|similar]] shapes that each include one of two legs instead of the hypotenuse (see [[#Similar figures on the three sides|Similar figures on the three sides]]).  In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself.  The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts.  Those two parts have the same shape as the original right triangle, and have the legs of the original triangle as their hypotenuses, and the sum of their areas is that of the original triangle.  Because the ratio of the area of a right triangle to the square of its hypotenuse is the same for similar triangles, the relationship between the areas of the three triangles holds for the squares of the sides of the large triangle as well.

===Algebraic proofs===
[[File:Pythagoras algebraic2.svg|thumb|upright|Diagram of the two algebraic proofs]]

The theorem can be proved algebraically using four copies of a right triangle with sides ''a'', ''b'' and ''c'', arranged inside a square with side ''c'' as in the top half of the diagram.<ref name=rotate>
{{cite web |url=http://www.cut-the-knot.org/pythagoras/index.shtml#3 |title=Cut-the-knot.org: Pythagorean theorem and its many proofs, Proof #3 |author=Alexander Bogomolny |date= |work=Cut the Knot |publisher= |accessdate=4 November 2010}}
</ref> The triangles are similar with area <math>\tfrac12ab</math>, while the small square has side {{nowrap|''b'' − ''a''}} and area {{nowrap|(''b'' − ''a'')<sup>2</sup>}}. The area of the large square is therefore

:<math>(b-a)^2+4\frac{ab}{2} = (b-a)^2+2ab = b^2-2ab+a^2+2ab = a^2+b^2. </math>

But this is a square with side ''c'' and area ''c''<sup>2</sup>, so

:<math>c^2 = a^2 + b^2. </math>

A similar proof uses four copies of the same triangle arranged symmetrically around a square with side ''c'', as shown in the lower part of the diagram.<ref>{{cite web |url=http://www.cut-the-knot.org/pythagoras/index.shtml#4 |title=Cut-the-knot.org: Pythagorean theorem and its many proofs, Proof #4 |author=Alexander Bogomolny |date= |work=Cut the Knot |publisher= |accessdate=4 November 2010}}</ref> This results in a larger square, with side {{nowrap|''a'' + ''b''}} and area {{nowrap|(''a'' + ''b'')<sup>2</sup>}}. The four triangles and the square side ''c'' must have the same area as the larger square,

:<math>(b+a)^2 = c^2 + 4\frac{ab}{2} = c^2+2ab,</math>

giving

:<math>c^2 = (b+a)^2 - 2ab = b^2+2ab+a^2-2ab = a^2 + b^2.</math>

[[File:Garfield Pythagoras.svg|thumb|upright|Diagram of Garfield's proof]]

A related proof was published by future U.S. President [[James A. Garfield]] (then a [[U.S. Representative]]) (see diagram).<ref name=Garfield>
Published in a weekly mathematics column: {{cite journal |journal=The New England Journal of Education |title=Pons Asinorum |volume=3 |issue=14 |page=161 |year=1876 |author=James A Garfield |ref=harv |url=http://www.maa.org/press/periodicals/convergence/mathematical-treasure-james-a-garfields-proof-of-the-pythagorean-theorem }} as noted in {{cite book |title=The mathematical universe: An alphabetical journey through the great proofs, problems, and personalities |url=https://books.google.com/?id=3tG_FRQ9N1QC&cd=1&dq=%22mathematical+universe%22+inauthor%3AWilliam+inauthor%3ADunham&q=New+England+Journal#search_anchor |author=William Dunham |year=1997 |publisher=Wiley |isbn=0-471-17661-3 |page=96 }} and in [http://www.math.usma.edu/people/rickey/hm/Dates/April.pdf A calendar of mathematical dates: April 1, 1876] {{Webarchive|url=https://web.archive.org/web/20100714153516/http://www.math.usma.edu/people/Rickey/hm/Dates/April.pdf |date=July 14, 2010 }} by V. Frederick Rickey
</ref><ref name=animation>{{cite web|title=Garfield's proof of the Pythagorean Theorem|first=David|last=Lantz|url=http://math.colgate.edu/faculty/dlantz/Pythpfs/Garfldpf.html|website=Math.Colgate.edu|accessdate=2018-01-14|archive-url=https://web.archive.org/web/20130828104818/http://math.colgate.edu/faculty/dlantz/Pythpfs/Garfldpf.html|archive-date=2013-08-28|url-status=dead}}</ref><ref>Maor, Eli, ''The Pythagorean Theorem'',  Princeton University Press, 2007: pp. 106-107.</ref> Instead of a square it uses a [[trapezoid]], which can be constructed from the square in the second of the above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown in the diagram. The [[trapezoid#Area|area of the trapezoid]] can be calculated to be half the area of the square, that is

:<math>\frac{1}{2}(b+a)^2.</math>

The inner square is similarly halved, and there are only two triangles so the proof proceeds as above except for a factor of <math>\frac{1}{2}</math>, which is removed by multiplying by two to give the result.

===Proof using differentials===
One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employing [[calculus]].<ref name="Staring">
{{cite journal |doi=10.2307/2691395 |title=The Pythagorean proposition: A proof by means of calculus |author=Mike Staring |journal=Mathematics Magazine |publisher=Mathematical Association of America |volume=69 |number=1 |pages=45–46 |year=1996 |jstor=2691395 |ref=harv}}
</ref><ref name=M_Hardy>
{{cite web |url=http://www.cut-the-knot.org/pythagoras |title=Pythagorean Theorem |accessdate=2010-05-09 |last=Bogomolny |first=Alexander |work=Interactive Mathematics Miscellany and Puzzles |publisher=Alexander Bogomolny |url-status=dead |archiveurl=https://web.archive.org/web/20100706200930/http://www.cut-the-knot.org/pythagoras/ |archivedate=2010-07-06 }}
</ref><ref>
{{cite journal |author= Bruce C. Berndt |title= Ramanujan – 100 years old (fashioned) or 100 years new (fangled)? |journal= The Mathematical Intelligencer |volume= 10 |pages= 24 |year= 1988 |doi=10.1007/BF03026638 |ref= harv |issue= 3}}
</ref>

The triangle ''ABC'' is a right triangle, as shown in the upper part of the diagram, with ''BC'' the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length ''y'', the side ''AC'' of length ''x'' and the side ''AB'' of length ''a'', as seen in the lower diagram part.

[[File:Pythag differential proof.svg|thumb|Diagram for differential proof]]
If ''x'' is increased by a small amount ''dx'' by extending the side ''AC'' slightly to ''D'', then ''y'' also increases by ''dy''. These form two sides of a triangle, ''CDE'', which (with ''E'' chosen so ''CE'' is perpendicular to the hypotenuse) is a right triangle approximately similar to ''ABC''. Therefore, the ratios of their sides must be the same, that is:

:<math> \frac{dy}{dx}=\frac xy.</math>
This can be rewritten as <math>y \, dy=x \, dx</math> , which is a [[differential equation]] that can be solved by direct integration:
:<math>\int y \, dy=\int x \, dx\,,</math>
giving
:<math>y^2=x^2+C.</math>
The constant can be deduced from ''x'' = 0, ''y'' = ''a'' to give the equation
:<math>y^2 = x^2 + a^2.</math>
This is more of an intuitive proof than a formal one: it can be made more rigorous if proper limits are used in place of ''dx'' and ''dy''.

==Converse==
The [[Theorem#Converse|converse]] of the theorem is also true:<ref name=Sally1>

{{cite book |title= Roots to Research |author1=Judith D. Sally |author2=Paul J. Sally Jr. |pages = 54–55 |chapter=Theorem 2.4 (Converse of the Pythagorean theorem). |url=https://books.google.com/books?id=nHxBw-WlECUC&pg=PA54 | publisher= [[American Mathematical Society]]
|isbn=0-8218-4403-2 |date=2007-12-21 }}
</ref>
<blockquote>For any three positive numbers ''a'', ''b'', and ''c'' such that {{nowrap|''a''<sup>2</sup> + ''b''<sup>2</sup> {{=}} ''c''<sup>2</sup>}}, there exists a triangle with sides ''a'', ''b'' and ''c'', and every such triangle has a right angle between the sides of lengths ''a'' and ''b''.</blockquote>

An alternative statement is:

<blockquote>For any triangle with sides ''a'', ''b'', ''c'', if {{nowrap|''a''<sup>2</sup> + ''b''<sup>2</sup> {{=}} ''c''<sup>2</sup>,}} then the angle between ''a'' and ''b'' measures 90°.</blockquote>

This converse also appears in Euclid's ''Elements'' (Book I, Proposition 48):<ref>
[http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI48.html Euclid's Elements, Book I, Proposition 48] From [http://aleph0.clarku.edu/~djoyce/java/elements/elements.html D.E. Joyce's web page] at Clark University
</ref> {{blockquote|text="If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right."}}

It can be proven using the [[law of cosines]] or as follows:

Let ''ABC'' be a triangle with side lengths ''a'', ''b'', and ''c'', with {{nowrap|''a''<sup>2</sup> + ''b''<sup>2</sup> {{=}} ''c''<sup>2</sup>.}} Construct a second triangle with sides of length ''a'' and ''b'' containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length ''c'' = {{radic|''a''<sup>2</sup> + ''b''<sup>2</sup>}}, the same as the hypotenuse of the first triangle. Since both triangles' sides are the same lengths ''a'', ''b'' and ''c'', the triangles are [[congruence (geometry)|congruent]] and must have the same angles. Therefore, the angle between the side of lengths ''a'' and ''b'' in the original triangle is a right angle.

The above proof of the converse makes use of the Pythagorean theorem itself. The converse can also be proven without assuming the Pythagorean theorem.<ref>Casey, Stephen, "The converse of the theorem of Pythagoras", ''[[Mathematical Gazette]]'' 92, July 2008, 309–313.</ref><ref>Mitchell, Douglas W., "Feedback on 92.47", ''Mathematical Gazette'' 93, March 2009, 156.</ref>

A [[corollary]] of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Let ''c'' be chosen to be the longest of the three sides and {{nowrap|''a'' + ''b'' > ''c''}} (otherwise there is no triangle according to the [[triangle inequality]]). The following statements apply:<ref name=Wilczynski>
{{cite book |title=Plane trigonometry and applications |author1=Ernest Julius Wilczynski |author2=[[Herbert Ellsworth Slaught]] |page=85 |url=https://books.google.com/?id=vxk3AAAAMAAJ&pg=PA85 |chapter=Theorem 1 and Theorem 2 |year=1914 |publisher=Allyn and Bacon}}
</ref>
* If {{nowrap|''a''<sup>2</sup> + ''b''<sup>2</sup> {{=}} ''c''<sup>2</sup>,}} then the [[Right triangle|triangle is right]].
* If {{nowrap|''a''<sup>2</sup> + ''b''<sup>2</sup> > ''c''<sup>2</sup>,}} then the [[Acute Triangle|triangle is acute]].
* If {{nowrap|''a''<sup>2</sup> + ''b''<sup>2</sup> < ''c''<sup>2</sup>,}} then the [[Acute and obtuse triangles|triangle is obtuse]].

[[Edsger W. Dijkstra]] has stated this proposition about acute, right, and obtuse triangles in this language:

:{{nowrap|sgn(''α'' + ''β'' − ''γ'') {{=}} sgn(''a''<sup>2</sup> + ''b''<sup>2</sup> − ''c''<sup>2</sup>),}}

where ''α'' is the angle opposite to side ''a'', ''β'' is the angle opposite to side ''b'', ''γ'' is the angle opposite to side ''c'', and sgn is the [[sign function]].<ref>{{cite web|url=http://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD975.html|title=On the theorem of Pythagoras|first= Edsger W.|last=Dijkstra|authorlink=Edsger W. Dijkstra|date=September 7, 1986|work=EWD975|publisher=E. W. Dijkstra Archive}}</ref>

==Consequences and uses of the theorem==

===Pythagorean triples===
{{Main article|Pythagorean triple}}
A Pythagorean triple has three positive integers ''a'', ''b'', and ''c'', such that {{nowrap|''a''<sup>2</sup> + ''b''<sup>2</sup> {{=}} ''c''<sup>2</sup>.}} In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths.<ref name=Sally0/> Such a triple is commonly written {{nowrap|(''a'', ''b'', ''c'').}} Some well-known examples are {{nowrap|(3, 4, 5)}} and {{nowrap|(5, 12, 13).}}

A primitive Pythagorean triple is one in which ''a'', ''b'' and ''c'' are [[coprime]] (the [[greatest common divisor]] of ''a'', ''b'' and ''c'' is 1).

The following is a list of primitive Pythagorean triples with values less than 100:
<!-- Please don't add NON-PRIMITIVE Pythagorean triples to this list such as (6,8,10) or (15,20,25) or (10,24,26). -->

:(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

===Reciprocal Pythagorean theorem===
Given a [[right triangle]] with sides <math>a,b,c</math> and [[Altitude (triangle)|altitude]] <math>d</math> (a line from the right angle and perpendicular to the [[hypotenuse]] <math>c</math>). The Pythagorean theorem has,
:<math>a^2+b^2 = c^2</math>
while the ''reciprocal Pythagorean theorem''<ref>R. B. Nelsen, Proof Without Words: A Reciprocal Pythagorean Theorem, Mathematics Magazine, 82, December 2009, p. 370</ref> or the ''upside down Pythagorean theorem''<ref>The upside-down Pythagorean theorem, Jennifer Richinick, The Mathematical Gazette, Vol. 92, No. 524 (July 2008), pp. 313-316</ref> relates the two [[Cathetus|legs]] <math>a,b</math> to  the altitude <math>d</math>,<ref>Alexander Bogomolny, Pythagorean Theorem for the Reciprocals,https://www.cut-the-knot.org/pythagoras/PTForReciprocals.shtml</ref>
:<math>\frac1{a^2}+\frac1{b^2} = \frac1{d^2}</math>
The equation can be transformed to,
:<math>\frac1{(xz)^2}+\frac1{(yz)^2} = \frac1{(xy)^2}</math>
where <math>x^2+y^2=z^2</math> for any non-zero [[Real number|real]] <math>x,y,z</math>. If the <math>a,b,d</math> are to be [[integer]]s, the smallest solution <math>a>b>d</math> is then 
:<math>\frac1{20^2}+\frac1{15^2} = \frac1{12^2}</math>
using the smallest Pythagorean triple <math>3,4,5</math>. The reciprocal Pythagorean theorem is a special case of the [[optic equation]]
:<math>\frac1{p}+\frac1{q} = \frac1{r}</math>
where the denominators are squares and also for a [[heptagonal triangle]] whose sides <math>p,q,r</math> are square numbers.

===Incommensurable lengths===
[[File:Euclid Corollary 5.svg|thumb|The [[spiral of Theodorus]]: A construction for line segments with lengths whose ratios are the square root of a positive integer]]
One of the consequences of the Pythagorean theorem is that line segments whose lengths are [[Commensurability (mathematics)|incommensurable]] (so the ratio of which is not a [[rational number]]) can be constructed using a [[Compass and straightedge constructions|straightedge and compass]]. Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the [[square root]] operation.

The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer.<ref name=Euclid_5>{{cite book |title=The Elements of Euclid: with many additional propositions, and explanatory notes, to which is prefixed an introductory essay on logic |first=Henry |last=Law |publisher=John Weale |year=1853 |url=https://books.google.com/?id=Ssb_OnVOGLgC&pg=PA49 |page=49 |chapter=Corollary 5 of Proposition XLVII (''Pythagoras's Theorem'')}}</ref> Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is not a perfect square, it is a realization of a length incommensurable with the unit, such as {{radical|2}}, {{radical|3}}, {{radical|5}}&nbsp;. For more detail, see [[Quadratic irrational]].

Incommensurable lengths conflicted with the Pythagorean school's concept of numbers as only whole numbers. The Pythagorean school dealt with proportions by comparison of integer multiples of a common subunit.<ref name=Lavine0>

{{cite book |title=Understanding the infinite |author=Shaughan Lavine |page=13 |url=https://books.google.com/books?id=GvGqRYifGpMC&pg=PA13 |isbn=0-674-92096-1 |year=1994 |publisher=Harvard University Press}}

</ref> According to one legend, [[Hippasus|Hippasus of Metapontum]] (''ca.'' 470&nbsp;B.C.) was drowned at sea for making known the existence of the irrational or incommensurable.<ref name=drown>

{{Harv|Heath|1921|loc= Vol I, pp. 65}}; Hippasus was on a voyage at the time, and his fellows cast him overboard. See {{cite journal |author= James R. Choike |title= The pentagram and the discovery of an irrational number |journal=The College Mathematics Journal |volume=11 |pages=312–316 |year=1980 |ref= harv}}

</ref><ref name=Fritz>
A careful discussion of Hippasus's contributions is found in
{{cite journal |title=The Discovery of Incommensurability by Hippasus of Metapontum |author=Kurt Von Fritz |journal=Annals of Mathematics | series = Second Series|volume= 46 |date=Apr 1945|pages= 242–264 |jstor=1969021 |issue= 2 |publisher=Annals of Mathematics |ref=harv |doi=10.2307/1969021}}

</ref>

===Complex numbers===
[[File:Complex conjugate picture.svg|right|thumb|The absolute value of a complex number ''z'' is the distance ''r'' from ''z'' to the origin]]
For any [[complex number]]
:<math>z = x + iy,</math>
the [[absolute value]] or modulus is given by
:<math>r = |z|=\sqrt{x^2 + y^2}.</math>
So the three quantities, ''r'', ''x'' and ''y'' are related by the Pythagorean equation,
:<math>r^2 = x^2 + y^2.</math>
Note that ''r'' is defined to be a positive number or zero but ''x'' and ''y'' can be negative as well as positive. Geometrically ''r'' is the distance of the ''z'' from zero or the origin ''O'' in the [[complex plane]].

This can be generalised to find the distance between two points, ''z''<sub>1</sub> and ''z''<sub>2</sub> say. The required distance is given by
:<math>|z_1 - z_2|=\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2},</math>
so again they are related by a version of the Pythagorean equation,
:<math>|z_1 - z_2|^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2.</math>

===Euclidean distance===
{{details|Euclidean distance}}
The distance formula in [[Cartesian coordinates]] is derived from the Pythagorean theorem.<ref name=Macdonald>{{cite book |title=Mastering algorithms with Perl |author1=Jon Orwant |author2=Jarkko Hietaniemi |author3=John Macdonald |url=https://books.google.com/books?id=z9xMfXGoWd0C&pg=PA426 |page=426 |chapter=Euclidean distance |isbn=1-56592-398-7 |year=1999 |publisher=O'Reilly Media, Inc}}</ref> If {{nowrap|(''x''<sub>1</sub>, ''y''<sub>1</sub>)}} and {{nowrap|(''x''<sub>2</sub>, ''y''<sub>2</sub>)}} are points in the plane, then the distance between them, also called the [[Euclidean distance]], is given by

:<math> \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}. </math>

More generally, in [[Euclidean space|Euclidean ''n''-space]], the Euclidean distance between two points, <math>A\,=\,(a_1,a_2,\dots,a_n)</math> and <math>B\,=\,(b_1,b_2,\dots,b_n)</math>, is defined, by generalization of the Pythagorean theorem, as:

:<math>\sqrt{(a_1-b_1)^2 + (a_2-b_2)^2 + \cdots + (a_n-b_n)^2} = \sqrt{\sum_{i=1}^n (a_i-b_i)^2}.</math>

If instead of Euclidean distance, the square of this value (the [[squared Euclidean distance]], or SED) is used, the resulting equation avoids square roots and is simply a sum of the SED of the coordinates:

:<math>(a_1-b_1)^2 + (a_2-b_2)^2 + \cdots + (a_n-b_n)^2 = \sum_{i=1}^n (a_i-b_i)^2.</math>

The squared form is a smooth, [[convex function]] of both points, and is widely used in [[optimization theory]] and [[statistics]], forming the basis of [[least squares]]. In [[information geometry]], more general notions of [[statistical distance]], known as [[divergence (statistics)|divergences]], are used, and the Pythagorean identity can be generalized to [[Bregman divergence]]s, allowing general forms of [[least squares]] to be used to solve non-linear problems.

===Euclidean distance in other coordinate systems===

If Cartesian coordinates are not used, for example, if [[polar coordinates]] are used in two dimensions or, in more general terms, if [[curvilinear coordinates]] are used, the formulas expressing the Euclidean distance are more complicated than the Pythagorean theorem, but can be derived from it. A typical example where the straight-line distance between two points is converted to curvilinear coordinates can be found in the [[Legendre polynomials#Applications of Legendre polynomials in physics|applications of Legendre polynomials in physics]]. The formulas can be discovered by using Pythagoras's theorem with the equations relating the curvilinear coordinates to Cartesian coordinates. For example, the polar coordinates {{nowrap|(''r'', ''θ'')}} can be introduced as:

:<math> x = r \cos \theta, \ y = r \sin \theta.</math>

Then two points with locations {{nowrap|(''r''<sub>1</sub>, ''θ''<sub>1</sub>)}} and {{nowrap|(''r''<sub>2</sub>, ''θ''<sub>2</sub>)}} are separated by a distance ''s'':

:<math>s^2 = (x_1 - x_2)^2 + (y_1-y_2)^2 = (r_1 \cos \theta_1 -r_2 \cos \theta_2 )^2 + (r_1 \sin \theta_1 -r_2 \sin \theta_2)^2.</math>
Performing the squares and combining terms, the Pythagorean formula for distance in Cartesian coordinates produces the separation in polar coordinates as:
:<math>\begin{align}s^2 &= r_1^2 +r_2^2 -2 r_1 r_2 \left( \cos \theta_1 \cos \theta_2 +\sin \theta_1 \sin \theta_2 \right)\\
 &= r_1^2 +r_2^2 -2 r_1 r_2 \cos \left( \theta_1 - \theta_2\right)\\
 &=r_1^2 +r_2^2 -2 r_1 r_2 \cos \Delta \theta, \end{align}</math>
using the trigonometric [[List of trigonometric identities#Product-to-sum and sum-to-product identities|product-to-sum formulas]]. This formula is the [[#Law of cosines|law of cosines]], sometimes called the generalized Pythagorean theorem.<ref>{{Cite book
|title=Plane Trigonometry and Tables
|first1=George
|last1=Wentworth
|publisher=BiblioBazaar, LLC
|year=2009
|isbn=1-103-07998-0
|page=116
|url=https://books.google.com/?id=Z-O57gUYmIgC
|postscript=<!--None-->}}, [https://books.google.com/books?id=Z-O57gUYmIgC&pg=PA116 Exercises, page 116]
</ref> From this result, for the case where the radii to the two locations are at right angles, the enclosed angle {{nowrap|Δ''θ'' {{=}} {{pi}}/2,}} and the form corresponding to Pythagoras's theorem is regained: <math>s^2 = r_1^2 + r_2^2.</math> The Pythagorean theorem, valid for right triangles, therefore is a special case of the more general law of cosines, valid for arbitrary triangles.

===Pythagorean trigonometric identity===
{{Main article|Pythagorean trigonometric identity}}
[[File:Trig functions.svg|thumb|Similar right triangles showing sine and cosine of angle θ]]
In a right triangle with sides ''a'', ''b'' and hypotenuse ''c'', [[trigonometry]] determines the [[sine]] and [[cosine]] of the angle ''θ'' between side ''a'' and the hypotenuse as:
:<math>\sin \theta = \frac{b}{c}, \quad \cos \theta = \frac{a}{c}.</math>

From that it follows:

:<math> {\cos}^2 \theta + {\sin}^2 \theta = \frac{a^2 + b^2}{c^2} = 1,</math>

where the last step applies Pythagoras's theorem. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity.<ref name= Leff>
{{cite book |title=PreCalculus the Easy Way |author=Lawrence S. Leff |url=https://books.google.com/?id=y_7yrqrHTb4C&pg=PA296 |page=296 |isbn=0-7641-2892-2 |edition=7th |publisher=Barron's Educational Series |year=2005}}
</ref> In similar triangles, the ratios of the sides are the same regardless of the size of the triangles, and depend upon the angles. Consequently, in the figure, the triangle with hypotenuse of unit size has opposite side of size sin ''θ'' and adjacent side of size cos ''θ'' in units of the hypotenuse.

===Relation to the cross product===
[[File:Cross product parallelogram.svg|right|thumb|The area of a parallelogram as a cross product; vectors '''a''' and '''b''' identify a plane and {{nowrap|'''a × b'''}} is normal to this plane.]]
The Pythagorean theorem relates the [[cross product]] and [[dot product]] in a similar way:<ref name=Massey>
{{cite journal |doi=10.2307/2323537 |title=Cross products of vectors in higher-dimensional Euclidean spaces |author=WS Massey |journal=The American Mathematical Monthly |volume=90 |date=Dec 1983 |pages=697–701 |jstor=2323537 |issue=10 |publisher=Mathematical Association of America |ref=harv}}
</ref>

:<math> \|\mathbf{a} \times \mathbf{b}\|^2 + (\mathbf{a} \cdot \mathbf{b})^2 = \|\mathbf{a}\|^2 \|\mathbf{b}\|^2.</math>

This can be seen from the definitions of the cross product and dot product, as

:<math>\begin{align} \mathbf{a} \times \mathbf{b} &= ab \mathbf{n} \sin{\theta} \\
 \mathbf{a} \cdot \mathbf{b} &= ab \cos{\theta}, \end{align}</math>

with '''n''' a unit vector normal to both '''a''' and '''b'''. The relationship follows from these definitions and the Pythagorean trigonometric identity.

This can also be used to define the cross product. By rearranging the following equation is obtained

:<math> \|\mathbf{a} \times \mathbf{b}\|^2 = \|\mathbf{a}\|^2 \|\mathbf{b}\|^2 - (\mathbf{a} \cdot \mathbf{b})^2.</math>

This can be considered as a condition on the cross product and so part of its definition, for example in [[seven-dimensional cross product|seven dimensions]].<ref name=n_dimensions>{{cite book |title=Clifford algebras and spinors |author=Pertti Lounesto |url=https://books.google.com/?id=kOsybQWDK4oC&pg=PA96 |page=96 |chapter=§7.4 Cross product of two vectors |isbn=0-521-00551-5 |year=2001 |publisher=Cambridge University Press |edition=2nd}}
</ref><ref name=Hildebrand>
{{cite book |title=Methods of applied mathematics |author=Francis Begnaud Hildebrand |page=24 |url=https://books.google.com/?id=17EZkWPz_eQC&pg=PA24|isbn=0-486-67002-3 |edition=Reprint of Prentice-Hall 1965 2nd|publisher=Courier Dover Publications |year=1992}}
</ref>

==Generalizations==

===Similar figures on the three sides===
A generalization of the Pythagorean theorem extending beyond the areas of squares on the three sides to [[similar figures]] was known by [[Hippocrates of Chios]] in the 5th century BC,<ref>Heath, T. L., ''A History of Greek Mathematics'', Oxford University Press, 1921; reprinted by Dover, 1981.</ref> and was included by [[Euclid]] in his ''[[Euclid's Elements|Elements]]'':<ref name=Euclid_VI>Euclid's ''Elements'': Book VI, Proposition VI 31: "In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle."</ref>

<blockquote>If one erects similar figures (see [[Euclidean geometry]]) with corresponding sides on the sides of a right triangle, then the sum of the areas of the ones on the two smaller sides equals the area of the one on the larger side.</blockquote>

This extension assumes that the sides of the original triangle are the corresponding sides of the three congruent figures (so the common ratios of sides between the similar figures are ''a:b:c'').<ref name=Putz>Putz, John F. and Sipka, Timothy A. "On generalizing the Pythagorean theorem", ''The College Mathematics Journal'' 34 (4), September 2003, pp. 291–295.</ref> While Euclid's proof only applied to convex polygons, the theorem also applies to concave polygons and even to similar figures that have curved boundaries (but still with part of a figure's boundary being the side of the original triangle).<ref name=Putz/>

The basic idea behind this generalization is that the area of a plane figure is [[Proportionality (mathematics)|proportional]] to the square of any linear dimension, and in particular is proportional to the square of the length of any side. Thus, if similar figures with areas ''A'', ''B'' and ''C'' are erected on sides with corresponding lengths ''a'', ''b'' and ''c'' then:

:<math>\frac{A}{a^2} = \frac{B}{b^2} = \frac{C}{c^2}\, ,</math>
:<math>\Rightarrow A + B = \frac{a^2}{c^2}C + \frac{b^2}{c^2}C\, .</math>

But, by the Pythagorean theorem, ''a''<sup>2</sup> + ''b''<sup>2</sup> = ''c''<sup>2</sup>, so ''A'' + ''B'' = ''C''.

Conversely, if we can prove that ''A'' + ''B'' = ''C'' for three similar figures without using the Pythagorean theorem, then we can work backwards to construct a proof of the theorem. For example, the starting center triangle can be replicated and used as a triangle ''C'' on its hypotenuse, and two similar right triangles (''A'' and ''B'' ) constructed on the other two sides, formed by dividing the central triangle by its [[Altitude (triangle)|altitude]]. The sum of the areas of the two smaller triangles therefore is that of the third, thus ''A'' + ''B'' = ''C'' and reversing the above logic leads to the Pythagorean theorem a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup>. (''See also [[#Einstein's proof by dissection without rearrangement|Einstein's proof by dissection without rearrangement]]'')
{|
| [[File:Pythagoras applied to similar triangles.svg|thumb|size=75%|Generalization for similar triangles,<br>green area {{nowrap|A + B {{=}} blue}} area C]]
| [[File:Pythagoras by similar triangles.svg|thumb|Pythagoras's theorem using similar right triangles]]
| [[File:Pythagoras by pentagons.svg|thumb|size=75%|Generalization for regular pentagons]]
|}

===Law of cosines===
[[File:Law of cosines2.svg|x200px|thumb|The separation ''s'' of two points {{nowrap|(r<sub>1</sub>, θ<sub>1</sub>)}} and {{nowrap|(r<sub>2</sub>, θ<sub>2</sub>)}} in [[polar coordinates]] is given by the [[law of cosines]]. Interior angle Δθ = θ<sub>1</sub>−θ<sub>2</sub>.]]
{{main article|Law of cosines}}
The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:<ref name= Leff1>

{{cite book |title=''cited work'' |author=Lawrence S. Leff |url=https://books.google.com/?id=y_7yrqrHTb4C&pg=PA326 |page=326 |isbn=0-7641-2892-2 |publisher=Barron's Educational Series |date=2005-05-01}}

</ref>
:: <math>a^2+b^2-2ab\cos{\theta}=c^2, </math>

where <math>\theta</math> is the angle between sides <math>a</math> and <math>b</math>.

When <math>\theta</math> is <math>\frac{\pi}{2}</math> radians or 90°, then <math>\cos{\theta} = 0</math>, and the formula reduces to the usual Pythagorean theorem.

===Arbitrary triangle===
[[File:Tâbit ibn Qorra modified.svg|thumb|Generalization of Pythagoras's theorem by [[Thābit ibn Qurra|Tâbit ibn Qorra]].<ref name=Eves>

{{cite book |title=Great moments in mathematics (before 1650) |author=Howard Whitley Eves |url=https://books.google.com/books?id=9_w5jDPTvCQC&pg=PA41 |chapter=§4.8:...generalization of Pythagorean theorem |isbn=0-88385-310-8 |year=1983 |page=41 |publisher=Mathematical Association of America}}

</ref> Lower panel: reflection of triangle CAD (top) to form triangle DAC, similar to triangle ABC (top).]]
At any selected angle of a general triangle of sides ''a, b, c'', inscribe an isosceles triangle such that the equal angles at its base θ are the same as the selected angle. Suppose the selected angle θ is opposite the side labeled ''c''. Inscribing the isosceles triangle forms triangle ''CAD'' with angle θ opposite side ''b'' and with side ''r'' along ''c''. A second triangle is formed with angle θ opposite side ''a'' and a side with length ''s'' along ''c'', as shown in the figure. [[Thābit ibn Qurra]] stated that the sides of the three triangles were related as:<ref name=Sayii>

{{cite journal |doi=10.1086/348837 |jstor=227603 |author=Aydin Sayili |date=Mar 1960 |title= Thâbit ibn Qurra's Generalization of the Pythagorean Theorem |journal= Isis |volume= 51 |issue=1 |pages= 35–37 |ref=harv}}</ref><ref name=Sally2>

{{cite book |title=Roots to Research: A Vertical Development of Mathematical Problems |author1=Judith D. Sally |author2=Paul Sally |chapter=Exercise 2.10 (ii) |page=62 |url=https://books.google.com/books?id=nHxBw-WlECUC&pg=PA62
|isbn=0-8218-4403-2 |date=2007-12-21 }}
</ref>
:<math> a^2 +b^2 =c(r+s) \ . </math>
As the angle θ approaches {{pi}}/2, the base of the isosceles triangle narrows, and lengths ''r'' and ''s'' overlap less and less. When θ = {{pi}}/2, ''ADB'' becomes a right triangle, ''r'' + ''s'' = ''c'', and the original Pythagorean theorem is regained.

One proof observes that triangle ''ABC'' has the same angles as triangle ''CAD'', but in opposite order. (The two triangles share the angle at vertex B, both contain the angle θ, and so also have the same third angle by the [[triangle postulate]].) Consequently, ''ABC'' is similar to the reflection of ''CAD'', the triangle ''DAC'' in the lower panel. Taking the ratio of sides opposite and adjacent to θ,
:<math>\frac{c}{b} = \frac{b}{r} \ .</math>
Likewise, for the reflection of the other triangle,
:<math>\frac{c}{a} = \frac{a}{s} \ . </math>
[[Clearing fractions]] and adding these two relations:
:<math> cs + cr = a^2 +b^2 \ , </math>
the required result.

The theorem remains valid if the angle <math> \theta </math> is obtuse so the lengths ''r'' and ''s'' are non-overlapping.

===General triangles using parallelograms===
[[File:Pythagoras for scalene triangle.svg|thumb|Generalization for arbitrary triangles,<br>green {{nowrap|area {{=}} blue}} area]]
[[File:Pythagoras construction.svg|thumb|Construction for proof of parallelogram generalization]]
[[Pappus's area theorem]] is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the original Pythagoras's theorem, and was considered a generalization by [[Pappus of Alexandria]] in 4&nbsp;AD<ref name=Jennings>For the details of such a construction, see {{cite book |title=Modern geometry with applications: with 150 figures |author=George Jennings |chapter=Figure 1.32: The generalized Pythagorean theorem |page=23 |url=https://books.google.com/?id=6OhcE7YQY8QC&pg=PA23 |isbn=0-387-94222-X |year=1997 |edition=3rd |publisher=Springer}}</ref><ref>Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. MAA, 2010, {{isbn|9780883853481}}, pp.&nbsp;77–78 ({{Google books|mIT5-BN_L0oC|excerpt|page=77}})</ref>

The lower figure shows the elements of the proof. Focus on the left side of the figure. The left green parallelogram has the same area as the left, blue portion of the bottom parallelogram because both have the same base ''b'' and height ''h''. However, the left green parallelogram also has the same area as the left green parallelogram of the upper figure, because they have the same base (the upper left side of the triangle) and the same height normal to that side of the triangle. Repeating the argument for the right side of the figure, the bottom parallelogram has the same area as the sum of the two green parallelograms.

===Solid geometry===
{{main article|Solid geometry}}
[[File:Pythagoras 3D.svg|thumb|Pythagoras's theorem in three dimensions relates the diagonal AD to the three sides.]]
[[File:Tetrahedron vertfig.svg|thumb|A tetrahedron with outward facing right-angle corner]]
In terms of solid geometry, Pythagoras's theorem can be applied to three dimensions as follows. Consider a rectangular solid as shown in the figure. The length of diagonal ''BD'' is found from Pythagoras's theorem as:

:<math> \overline{BD}^{\,2} = \overline{BC}^{\,2} + \overline{CD}^{\,2} \ ,</math>

where these three sides form a right triangle. Using horizontal diagonal ''BD'' and the vertical edge ''AB'', the length of diagonal ''AD'' then is found by a second application of Pythagoras's theorem as:

:<math> \overline{AD}^{\,2} = \overline{AB}^{\,2} + \overline{BD}^{\,2} \ ,</math>
or, doing it all in one step:
:<math> \overline{AD}^{\,2} = \overline{AB}^{\,2} + \overline{BC}^{\,2} + \overline{CD}^{\,2} \ .</math>

This result is the three-dimensional expression for the magnitude of a vector '''v''' (the diagonal AD) in terms of its orthogonal components {'''v'''<sub>k</sub>} (the three mutually perpendicular sides):

:<math>\|\mathbf{v}\|^2 = \sum_{k=1}^3 \|\mathbf{v}_k\|^2.</math>

This one-step formulation may be viewed as a generalization of Pythagoras's theorem to higher dimensions. However, this result is really just the repeated application of the original Pythagoras's theorem to a succession of right triangles in a sequence of orthogonal planes.

A substantial generalization of the Pythagorean theorem to three dimensions is [[de Gua's theorem]], named for [[Jean Paul de Gua de Malves]]: If a [[tetrahedron]] has a right angle corner (like a corner of a [[cube (geometry)|cube]]), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. This result can be generalized as in the "''n''-dimensional Pythagorean theorem":<ref name=Bhatia>

{{cite book |title=Matrix analysis |author=Rajendra Bhatia |url=https://books.google.com/?id=eay3HALl620C&pg=PA21 |page=21 |isbn=0-387-94846-5 |year=1997 |publisher=Springer}}

</ref>
{{Blockquote|text=Let <math>x_1, x_2, \ldots,x_n </math> be orthogonal vectors in ℝ<sup>n</sup>. Consider the ''n''-dimensional simplex ''S'' with vertices <math>0,x_1,\ldots, x_n</math>. (Think of the (''n''&nbsp;−&nbsp;1)-dimensional simplex with vertices <math>x_1,\ldots,x_n</math> not including the origin as the "hypotenuse" of ''S'' and the remaining (''n''&nbsp;−&nbsp;1)-dimensional faces of ''S'' as its "legs".) Then the square of the volume of the hypotenuse of ''S'' is the sum of the squares of the volumes of the ''n'' legs.}}

This statement is illustrated in three dimensions by the tetrahedron in the figure. The "hypotenuse" is the base of the tetrahedron at the back of the figure, and the "legs" are the three sides emanating from the vertex in the foreground. As the depth of the base from the vertex increases, the area of the "legs" increases, while that of the base is fixed. The theorem suggests that when this depth is at the value creating a right vertex, the generalization of Pythagoras's theorem applies. In a different wording:<ref name=Wong>
For an extended discussion of this generalization, see, for example, [http://www.dpmms.cam.ac.uk/~ww278/papers/gp.pdf Willie W. Wong] {{Webarchive|url=https://web.archive.org/web/20091229111557/http://www.dpmms.cam.ac.uk/~ww278/papers/gp.pdf |date=2009-12-29 }} 2002, ''A generalized n-dimensional Pythagorean theorem''.
</ref>
{{blockquote|text=Given an ''n''-rectangular ''n''-dimensional simplex, the square of the (''n''&nbsp;−&nbsp;1)-content of the [[Facet (mathematics)|facet]] opposing the right vertex will equal the sum of the squares of the (''n''&nbsp;−&nbsp;1)-contents of the remaining facets.}}

===Inner product spaces===
{{See also|Hilbert space}}
[[File:Parallelogram equality.svg|thumb|Vectors involved in the parallelogram law]]
The Pythagorean theorem can be generalized to [[inner product space]]s,<ref name=van_der_Heijden>

{{cite book |title=Classification, parameter estimation, and state estimation |author1=Ferdinand van der Heijden |author2=Dick de Ridder |url=https://books.google.com/books?id=krSB9PIKMSYC&pg=PA357 |page=357 |isbn=0-470-09013-8 |year=2004 |publisher=Wiley}}

</ref> which are generalizations of the familiar 2-dimensional and 3-dimensional [[Euclidean spaces]]. For example, a [[Function (mathematics)|function]] may be considered as a [[vector space|vector]] with infinitely many components in an inner product space, as in [[functional analysis]].<ref name=Lin>

{{cite book |title=Finite element methods: accuracy and improvement |author1=Qun Lin |author2=Jiafu Lin |url=https://books.google.com/books?id=cMvAqzMuAWgC&pg=PA23 |page=23 |isbn=7-03-016656-6 |year=2006 |publisher=Elsevier}}

</ref>

In an inner product space, the concept of [[perpendicular]]ity is replaced by the concept of [[orthogonal]]ity: two vectors '''v''' and '''w''' are orthogonal if their inner product <math> \langle \mathbf{v} , \mathbf{w}\rangle </math> is zero. The [[Inner product space#Definition|inner product]] is a generalization of the [[dot product]] of vectors. The dot product is called the ''standard'' inner product or the ''Euclidean'' inner product. However, other inner products are possible.<ref name=Rorres>

{{cite book |title=Elementary Linear Algebra: Applications Version |author1=Howard Anton |author2=Chris Rorres |url=https://books.google.com/books?id=1PJ-WHepeBsC&pg=PA336 |page=336 |isbn=0-470-43205-5 |year=2010 |edition=10th |publisher=Wiley}}

</ref>

The concept of length is replaced by the concept of the [[normed vector space|norm]] ||'''v'''|| of a vector '''v''', defined as:<ref name=Saxe>

{{cite book |title=Beginning functional analysis |author=Karen Saxe|authorlink= Karen Saxe |url=https://books.google.com/books?id=QALoZC64ea0C&pg=PA7 |page=7 |chapter=Theorem 1.2 |isbn=0-387-95224-1 |year=2002 |publisher=Springer}}

</ref>
:<math>\lVert \mathbf{v} \rVert \equiv \sqrt{\langle \mathbf{v},\mathbf{v}\rangle} \, .</math>

In an inner-product space, the '''Pythagorean theorem''' states that for any two orthogonal vectors '''v''' and '''w''' we have
:<math>\left\| \mathbf{v} + \mathbf{w} \right\|^2 = \left\| \mathbf{v} \right\|^2 + \left\| \mathbf{w} \right\|^2 .</math>
Here the vectors '''v''' and '''w''' are akin to the sides of a right triangle with hypotenuse given by the [[Euclidean vector#Addition and subtraction|vector sum]] '''v'''&nbsp;+&nbsp;'''w'''. This form of the Pythagorean theorem is a consequence of the [[Inner product space#Definition|properties of the inner product]]:
:<math>\left\| \mathbf{v} + \mathbf{w} \right\|^2 =\langle \mathbf{ v+w},\ \mathbf{ v+w}\rangle = \langle \mathbf{ v},\ \mathbf{ v}\rangle +\langle \mathbf{ w},\ \mathbf{ w}\rangle +\langle\mathbf{ v,\ w }\rangle + \langle\mathbf{ w,\ v }\rangle \ = \left\| \mathbf{v}\right\|^2 + \left\| \mathbf{w}\right\|^2, </math>
where the inner products of the cross terms are zero, because of orthogonality.

A further generalization of the Pythagorean theorem in an inner product space to non-orthogonal vectors is the ''[[Parallelogram equality#Normed vector spaces satisfying the parallelogram law|parallelogram law]]'' :<ref name=Saxe/>

:<math>2\|\mathbf v\|^2 +2 \|\mathbf w\|^2 = \|\mathbf {v + w} \|^2 +\| \mathbf{v-w}\|^2 \ , </math>

which says that twice the sum of the squares of the lengths of the sides of a parallelogram is the sum of the squares of the lengths of the diagonals. Any norm that satisfies this equality is ''[[ipso facto]]'' a norm corresponding to an inner product.<ref name=Saxe/>

{{anchor|Decomposition}}
The Pythagorean identity can be extended to sums of more than two orthogonal vectors. If '''v'''<sub>''1''</sub>, '''v'''<sub>''2''</sub>, ..., '''v'''<sub>''n''</sub> are pairwise-orthogonal vectors in an inner-product space, then application of the Pythagorean theorem to successive pairs of these vectors (as described for 3-dimensions in the section on [[#Solid geometry|solid geometry]]) results in the equation<ref name='Douglas1998'>{{cite book | last = Douglas | first = Ronald G. | title = Banach Algebra Techniques in Operator Theory, 2nd edition | publisher = Springer-Verlag New York, Inc | year = 1998 | location = New York, New York | pages = 60–61 | url = https://books.google.com/?id=-OdfXeNmrT0C&pg=PA60#v=onepage&q | isbn = 978-0-387-98377-6}}</ref>
:<math>\left\|\sum_{k=1}^n\mathbf{v}_k\right\|^2=\sum_{k=1}^n\|\mathbf{v}_k\|^2</math>

===Sets of ''m''-dimensional objects in ''n''-dimensional space===

Another generalization of the Pythagorean theorem applies to [[Lebesgue-measurable]] sets of objects in any number of dimensions.  Specifically, the square of the measure of an ''m''-dimensional set of objects in one or more parallel ''m''-dimensional [[Flat (geometry)|flats]] in ''n''-dimensional [[Euclidean space]] is equal to the sum of the squares of the measures of the [[Orthogonality|orthogonal]] projections of the object(s) onto all ''m''-dimensional coordinate subspaces.<ref name=Conant>
{{cite journal |doi=10.2307/2319528 |title=Generalized Pythagorean Theorem |author1=Donald R Conant  |author2=William A Beyer  |lastauthoramp=yes |journal=The American Mathematical Monthly |volume=81 |date=Mar 1974 |pages=262–265 |jstor=2319528 |issue=3 |publisher=Mathematical Association of America |ref=harv}}
</ref>

In mathematical terms:
:<math>\mu^2_{ms} = \sum_{i=1}^{x}\mathbf{\mu^2}_{mp_i}</math>
where:
* <math>\mu_m</math> is a measure in ''m''-dimensions (a length in one dimension, an area in two dimensions, a volume in three dimensions, etc.).
* <math>s</math> is a set of one or more non-overlapping ''m''-dimensional objects in one or more parallel ''m''-dimensional flats in ''n''-dimensional Euclidean space.
* <math>\mu_{ms}</math> is the total measure (sum) of the set of ''m''-dimensional objects.
* <math>p</math> represents an ''m''-dimensional projection of the original set onto an orthogonal coordinate subspace.
* <math>\mu_{mp_i}</math> is the measure of the ''m''-dimensional set projection onto ''m''-dimensional coordinate subspace <math>i</math>.  Because object projections can overlap on a coordinate subspace, the measure of each object projection in the set must be calculated individually, then measures of all projections added together to provide the total measure for the set of projections on the given coordinate subspace.
* <math>x</math> is the number of orthogonal, ''m''-dimensional coordinate subspaces in ''n''-dimensional space ({{math|'''R'''<sup>''n''</sup>}}) onto which the ''m''-dimensional objects are projected (''m'' ≤ ''n''):
:: <math>x = \binom{n}{m} = \frac{n!}{m!(n-m)!}</math>

===Non-Euclidean geometry===
{{main article|Non-Euclidean geometry}}
{{See also|Hilbert's axioms}}
The Pythagorean theorem is derived from the axioms of [[Euclidean geometry]], and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theorem [[Parallel postulate#Equivalent properties|implies, and is implied by, Euclid's Parallel (Fifth) Postulate]].<ref name=Parallel>

{{cite book |title=CRC concise encyclopedia of mathematics |author= Eric W. Weisstein |url=https://books.google.com/books?id=aFDWuZZslUUC&pg=PA2147 |page=2147 |quote=The parallel postulate is equivalent to the ''Equidistance postulate'', ''Playfair axiom'', ''Proclus axiom'', the ''Triangle postulate'' and the ''Pythagorean theorem''. |edition=2nd |isbn=1-58488-347-2 |year=2003}}

</ref><ref name= Pruss>

{{cite book |title=The principle of sufficient reason: a reassessment |author= Alexander R. Pruss |quote=We could include...the parallel postulate and derive the Pythagorean theorem. Or we could instead make the Pythagorean theorem among the other axioms and derive the parallel postulate. |isbn=0-521-85959-X |year=2006 |publisher=Cambridge University Press |page=11 |url=https://books.google.com/books?id=8qAxk1rXIjQC&pg=PA11}}

</ref> Thus, right triangles in a [[non-Euclidean geometry]]<ref name=false>

{{cite book |title=''cited work'' |author=Stephen W. Hawking |page=4 |url=https://books.google.com/books?id=3zdFSOS3f4AC&pg=PA4 |isbn=0-7624-1922-9 |year=2005}}

</ref>
do not satisfy the Pythagorean theorem. For example, in [[spherical geometry]], all three sides of the right triangle (say ''a'', ''b'', and ''c'') bounding an octant of the unit sphere have length equal to {{pi}}/2, and all its angles are right angles, which violates the Pythagorean theorem because 
<math> a^2 + b^2 = 2 c^2 > c^2 </math>. 

Here two cases of non-Euclidean geometry are considered—[[spherical geometry]] and [[hyperbolic geometry|hyperbolic plane geometry]]; in each case, as in the Euclidean case for non-right triangles, the result replacing the Pythagorean theorem follows from the appropriate law of cosines.

However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition that the triangle be right is replaced with the condition that two of the angles sum to the third, say ''A''+''B'' = ''C''. The sides are then related as follows: the sum of the areas of the circles with diameters ''a'' and ''b'' equals the area of the circle with diameter ''c''.<ref>{{cite journal|author=Victor Pambuccian|title=Maria Teresa Calapso's Hyperbolic Pythagorean Theorem|journal=The Mathematical Intelligencer|volume=32|date=December 2010|page =2|doi=10.1007/s00283-010-9169-0|issue=4}}</ref>

====Spherical geometry====
{{main article|Spherical geometry}}
[[File:Triangle sphérique.svg|thumb|Spherical triangle]]
For any right triangle on a sphere of radius ''R'' (for example, if γ in the figure is a right angle), with sides ''a'', ''b'', ''c'', the relation between the sides takes the form:<ref name=O>
{{cite book |title=Elementary differential geometry |author=Barrett O'Neill |url=https://books.google.com/?id=OtbNXAIve_AC&pg=PA441 |chapter=Exercise 4 |page=441 |isbn=0-12-088735-5 |year=2006 |edition=2nd |publisher=Academic Press}}
</ref>

:<math> \cos \left(\frac{c}{R}\right)=\cos \left(\frac{a}{R}\right)\cos \left(\frac{b}{R}\right).</math>

This equation can be derived as a special case of the [[spherical law of cosines]] that applies to all spherical triangles:

:<math> \cos \left(\frac{c}{R}\right)=\cos \left(\frac{a}{R}\right)\cos \left(\frac{b}{R}\right) +\sin\left(\frac{a}{R}\right) \sin\left(\frac{b}{R}\right) \cos \gamma \ .</math>

By expressing the [[Maclaurin series]] for the cosine function as an [[asymptotic expansion]] with the remainder term in [[big O notation]],

:<math> \cos x = 1 - \frac{x^2}{2} + O(x^4) \text{ as } x \to 0\ ,</math>

it can be shown that as the radius ''R'' approaches infinity and the arguments ''a/R'', ''b/R'', and ''c/R'' tend to zero, the spherical relation between the sides of a right triangle approaches the Euclidean form of the Pythagorean theorem. Substituting the asymptotic expansion for each of the cosines into the spherical relation for a right triangle yields

:<math>1-\frac{1}{2}\left(\frac{c}{R}\right)^2 + O\left(\frac{1}{R^4}\right) = \left[1-\frac{1}{2}\left(\frac{a}{R}\right)^2 + O\left(\frac{1}{R^4}\right) \right]\left[1-\frac{1}{2}\left(\frac{b}{R}\right)^2 + O\left(\frac{1}{R^4}\right) \right] \text{ as }R\to\infty\ .</math>

The constants ''a''<sup>4</sup>, ''b''<sup>4</sup>, and ''c''<sup>4</sup> have been absorbed into the big ''O'' remainder terms since they are independent of the radius ''R''. This asymptotic relationship can be further simplified by multiplying out the bracketed quantities, cancelling the ones, multiplying through by −2, and collecting all the error terms together:

:<math>\left(\frac{c}{R}\right)^2 = \left(\frac{a}{R}\right)^2 + \left(\frac{b}{R}\right)^2 + O\left(\frac{1}{R^4}\right)\text{ as }R\to\infty\ .</math>

After multiplying through by ''R''<sup>2</sup>, the Euclidean Pythagorean relationship ''c''<sup>2</sup> = ''a''<sup>2</sup> + ''b''<sup>2</sup> is recovered in the limit as the radius ''R'' approaches infinity (since the remainder term tends to zero):

:<math>c^2= a^2 + b^2 + O\left(\frac{1}{R^2}\right)\text{ as }R\to\infty\ .</math>

For small right triangles (''a'', ''b'' << ''R''), the cosines can be eliminated to avoid [[loss of significance]], giving
:<math> \sin^2 \frac{c}{2R} = \sin^2 \frac{a}{2R} + \sin^2 \frac{b}{2R} - 2 \sin^2 \frac{a}{2R} \sin^2 \frac{b}{2R} \,.</math>

====Hyperbolic geometry====
{{main article|Hyperbolic geometry}}
{{See also|Hyperbolic triangle |Gaussian curvature}}
[[File:Hyperbolic triangle.svg|thumb|[[Hyperbolic triangle]] ]]
In a hyperbolic space with uniform curvature −1/''R''<sup>2</sup>, for a right triangle with legs ''a'', ''b'', and hypotenuse ''c'', the relation between the sides takes the form:<ref name=Stahl>

{{cite book |title=The Poincaré half-plane: a gateway to modern geometry |author=Saul Stahl |page=122 |url=https://books.google.com/?id=TABicHVMQhMC&pg=PA122 |chapter=Theorem 8.3| isbn=0-86720-298-X |year=1993 |publisher=Jones & Bartlett Learning}}

</ref>

:<math> \cosh \frac{c}{R} = \cosh \frac{a}{R} \, \cosh \frac{b}{R} </math>

where cosh is the [[hyperbolic function|hyperbolic cosine]]. This formula is a special form of the [[hyperbolic law of cosines]] that applies to all hyperbolic triangles:<ref name=Gilman>

{{cite book |title=Two-generator discrete subgroups of PSL(2,R) |author=Jane Gilman|authorlink=Jane Piore Gilman |url=https://books.google.com/?id=YRFz9Zj_vAAC&pg=PA74 |chapter=Hyperbolic triangles |isbn=0-8218-0361-1 |year=1995 |publisher=American Mathematical Society Bookstore}}

</ref>

:<math>\cosh \frac{c}{R} = \cosh \frac{a}{R} \ \cosh \frac{b}{R} - \sinh \frac{a}{R} \ \sinh \frac{b}{R} \ \cos \gamma \ , </math>

with γ the angle at the vertex opposite the side ''c''.

By using the [[Maclaurin series]] for the hyperbolic cosine, {{nowrap|cosh ''x'' ≈ 1 + ''x''<sup>2</sup>/2}}, it can be shown that as a hyperbolic triangle becomes very small (that is, as ''a'', ''b'', and ''c'' all approach zero), the hyperbolic relation for a right triangle approaches the form of Pythagoras's theorem.

For small right triangles (''a'', ''b'' << ''R''), the hyperbolic cosines can be eliminated to avoid [[loss of significance]], giving
:<math> \sinh^2 \frac{c}{2R} = \sinh^2 \frac{a}{2R} + \sinh^2 \frac{b}{2R} + 2 \sinh^2 \frac{a}{2R} \sinh^2 \frac{b}{2R} \,.</math>

====Very small triangles====
For any uniform curvature ''K'' (positive, zero, or negative), in very small right triangles (|''K''|''a''<sup>2</sup>, |''K''|''b''<sup>2</sup> << 1) with hypotenuse ''c'', it can be shown that
:<math> c^2 = a^2 + b^2 - \frac{K}{3} a^2 b^2 - \frac{K^2}{45} a^2 b^2 (a^2 + b^2) - \frac{2 K^3}{945} a^2 b^2 (a^2 - b^2)^2 + O (K^4 c^{10}) \,.</math>

===Differential geometry===
{{main article|Differential geometry}}
[[File:Line elements2.svg|thumb|Distance between infinitesimally separated points in [[Cartesian coordinates]] (top) and [[polar coordinates]] (bottom), as given by Pythagoras's theorem]]
On an infinitesimal level, in three dimensional space, Pythagoras's theorem describes the distance between two infinitesimally separated points as:

:<math>ds^2 = dx^2 + dy^2 +dz^2,</math>

with ''ds'' the element of distance and (''dx'', ''dy'', ''dz'') the components of the vector separating the two points. Such a space is called a [[Euclidean space]]. However, in [[Riemannian geometry]], a generalization of this expression useful for general coordinates (not just Cartesian) and general spaces (not just Euclidean) takes the form:<ref name=Chow>
{{cite book |title=Mathematical methods for physicists: a concise introduction |author=Tai L. Chow |page=52 |url=https://books.google.com/?id=MpRXPOYZzfUC&pg=PA52 |isbn=0-521-65544-7 |year=2000 |publisher=Cambridge University Press}}
</ref>

:<math>ds^2 = \sum_{i,j}^n g_{ij}\, dx_i\, dx_j </math>

which is called the [[Metric tensor#Arclength and the line element|metric tensor]]. (Sometimes, by abuse of language, the same term is applied to the set of coefficients {{nowrap|''g''<sub>''ij''</sub>}}.) It may be a function of position, and often describes [[curved space]]. A simple example is Euclidean (flat) space expressed in [[curvilinear coordinates]]. For example, in [[polar coordinates]]:

:<math>ds^2 = dr^2 + r^2 d\theta^2 \ . </math>

==History==
[[File:Plimpton 322.jpg|thumb|The [[Plimpton 322|Plimpton 322 tablet]] records Pythagorean triples from Babylonian times.<ref name=Neugebauer>{{harvnb|Neugebauer|1969|p=36}}.</ref>]]
There is debate whether the Pythagorean theorem was discovered once, or many times in many places, and the date of first discovery is uncertain, as is the date of the first proof.  Historians of Mesopotamian mathematics have concluded that the Pythagorean rule was in widespread use during the [[First Babylonian dynasty|Old Babylonian period]] (20th to 16th centuries BC), over a thousand years before Pythagoras was born.<ref>{{harvnb|Neugebauer|1969}}: p. 36 "In other words it was known during the whole duration of Babylonian mathematics that the sum of the squares on the lengths of the sides of a right triangle equals the square of the length of the hypotenuse."</ref><ref>{{cite journal|author=Friberg, Joran|url=https://www.researchgate.net/profile/Joeran_Friberg/publication/222892801_Methods_and_traditions_of_Babylonian_mathematics_Plimpton_322_Pythagorean_Triples_and_the_Babylonian_Triangle_Parameter_Equations/links/0deec52810c5445463000000.pdf|title=Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations|journal= Historia Mathematica|volume=8|pages=277&ndash;318|year=1981|doi=10.1016/0315-0860(81)90069-0}}: p. 306 "Although Plimpton 322 is a unique text of its kind, there are several other known texts testifying that the Pythagorean theorem was well known to the mathematicians of the Old Babylonian period."</ref><ref>{{cite conference | last=Høyrup | first=Jens | authorlink=Jens Høyrup| contribution=Pythagorean ‘Rule’ and ‘Theorem’ – Mirror of the Relation Between Babylonian and Greek Mathematics | pages=393–407 | editor-last=Renger | editor-first=Johannes | title=Babylon: Focus mesopotamischer Geschichte, Wiege früher Gelehrsamkeit, Mythos in der Moderne. 2. Internationales Colloquium der Deutschen Orient-Gesellschaft 24.–26. März 1998 in Berlin | publisher=Berlin: Deutsche Orient-Gesellschaft / Saarbrücken: SDV Saarbrücker Druckerei und Verlag | url=http://akira.ruc.dk/~jensh/Publications/Pythrule.pdf}}, p. 406, "''To judge from this evidence alone'' it is therefore likely that the Pythagorean rule was discovered within the lay surveyors’ environment, possibly as a spin-off from the problem treated in Db<sub>2</sub>-146, somewhere between 2300 and 1825 BC."  ([[Db2-146|Db<sub>2</sub>-146]] is an Old Babylonian clay tablet from [[Eshnunna]] concerning the computation of the sides of a rectangle given its area and diagonal.)</ref><ref>{{cite book |authorlink=Eleanor Robson |last=Robson |first=E. |title=Mathematics in Ancient Iraq: A Social History |location= |publisher=Princeton University Press |year=2008 }}: p. 109 "Many Old Babylonian mathematical practitioners … knew that the square on the diagonal of a right triangle had the same area as the sum of the squares on the length and width: that relationship is used in the worked solutions to word problems on cut-and-paste ‘algebra’ on seven different tablets, from Ešnuna, Sippar, Susa, and an unknown location in southern Babylonia."</ref> The history of the theorem can be divided into four parts: knowledge of [[Pythagorean triples]], knowledge of the relationship among the sides of a [[right triangle]], knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system.

Written between 2000 and 1786&nbsp;BC, the [[Middle Kingdom of Egypt|Middle Kingdom]] [[Egypt]]ian ''[[Berlin Papyrus 6619]]'' includes a problem whose solution is the [[Pythagorean triple]] 6:8:10, but the problem does not mention a triangle. The [[Mesopotamia]]n tablet ''[[Plimpton 322]]'', written between 1790 and 1750&nbsp;BC during the reign of [[Hammurabi]] the Great, contains many entries closely related to Pythagorean triples.

In [[India]], the ''[[Baudhayana]] [[Sulba Sutras|Sulba Sutra]]'', the dates of which are given variously as between the 8th and 5th century BC,<ref>{{cite book|author=Kim Plofker |title=Mathematics in India |url=https://books.google.com/books?id=DHvThPNp9yMC |year=2009|publisher=Princeton University Press |isbn=0-691-12067-6 |pages=17–18, with footnote 13 for Sutra identical to the Pythagorean theorem}}</ref> contains a list of [[Pythagorean triples]] discovered algebraically, a statement of the Pythagorean theorem, and a [[geometry|geometrical]] proof of the Pythagorean theorem for an [[isosceles]] right triangle.
The ''[[Apastamba]] Sulba Sutra'' (c. 600&nbsp;BC) contains a numerical proof of the general Pythagorean theorem, using an area computation. Van der Waerden believed that "it was certainly based on earlier traditions". Carl Boyer states that the Pythagorean theorem in ''[[Shulba Sutras|Śulba-sũtram]]'' may have been influenced by ancient Mesopotamian math, but there is no conclusive evidence in favor or opposition of this possibility.<ref>
{{cite book|author1=Carl Benjamin Boyer |author2=Uta C. Merzbach|author2-link=Uta Merzbach |authorlink=Carl Benjamin Boyer|year=2011 |title=A history of mathematics, 3rd Edition |url=https://books.google.com/books?id=bR9HAAAAQBAJ |chapter=China and India| page=229| isbn= 978-0470525487 |quote='''Quote:''' [In Sulba-sutras,] we find rules for the construction of right angles by means of triples of cords the lengths of which form Pythagorean triages, such as 3, 4, and 5, or 5, 12, and 13, or 8, 15, and 17, or 12, 35, and 37. Although Mesopotamian influence in the ''Sulvasũtras'' is not unlikely, we know of no conclusive evidence for or against this. Aspastamba knew that the square on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent sides. Less easily explained is another rule given by Apastamba – one that strongly resembles some of the geometric algebra in Book II of Euclid's ''Elements''. (...)|publisher=Wiley}}</ref>

[[File:Chinese pythagoras.jpg|thumb|250px|Geometric proof of the Pythagorean theorem from the ''[[Zhoubi Suanjing]]''.]]
With contents known much earlier, but in surviving texts dating from roughly the 1st century BC, the [[China|Chinese]] text ''[[Zhoubi Suanjing]]'' (周髀算经), (''The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven'') gives a reasoning for the Pythagorean theorem for the (3, 4, 5) triangle—in China it is called the "Gougu theorem" (勾股定理).<ref name=Crease>
{{cite book |title=The great equations: breakthroughs in science from Pythagoras to Heisenberg |page=25 |author=Robert P. Crease |isbn=0-393-06204-X |year=2008 |publisher=W W Norton & Co.}}
</ref><ref name=Cullen>

A rather extensive discussion of the origins of the various texts in the Zhou Bi is provided by {{cite book |title=Astronomy and Mathematics in Ancient China: The 'Zhou Bi Suan Jing' |author=Christopher Cullen |url=https://books.google.com/?id=U9E88abLP10C&pg=PA139&dq=to+datable++events+%22relate+the+material%22&cd=1#v=onepage&q=to%20datable%20%20events%20%22relate%20the%20material%22 |pages=139 ''ff'' |isbn=0-521-03537-6 |year=2007 |publisher=Cambridge University Press }}

</ref> During the [[Han Dynasty]] (202&nbsp;BC to 220&nbsp;AD), Pythagorean triples appear in ''[[The Nine Chapters on the Mathematical Art]]'',<ref name=Hui>

This work is a compilation of 246 problems, some of which survived the book burning of 213&nbsp;BC, and was put in final form before 100&nbsp;AD. It was extensively commented upon by Liu Hui in 263&nbsp;AD. {{cite book |title=Sherlock Holmes in Babylon: and other tales of mathematical history |author=Philip D Straffin, Jr. |chapter=Liu Hui and the first golden age of Chinese mathematics |editor1=Marlow Anderson |editor2=Victor J. Katz |editor3=Robin J. Wilson |url=https://books.google.com/books?id=BKRE5AjRM3AC&pg=PA69 |pages=69 ''ff'' |isbn=0-88385-546-1 |year=2004 |publisher=Mathematical Association of America}} See particularly §3: ''Nine chapters on the mathematical art'', pp. 71 ''ff''.

</ref> together with a mention of right triangles.<ref name=Shen>

{{cite book |title=The nine chapters on the mathematical art: companion and commentary |author1=Kangshen Shen |author2=John N. Crossley |author3=Anthony Wah-Cheung Lun |page=488 |url=https://books.google.com/?id=eiTJHRGTG6YC&pg=PA488 |isbn=0-19-853936-3 |year=1999 |publisher=Oxford University Press}}

</ref> Some believe the theorem arose first in [[China]],<ref name=Centaurus>

In particular, Li Jimin; see {{cite book |title=Centaurus, Volume 39 |url=https://books.google.com/?id=UJlFAAAAYAAJ&q=%22Shang+Gao+Theorem%22&dq=%22Shang+Gao+Theorem%22&cd=2 |pages=193, 205 |year=1997 |publisher=Munksgaard|location=Copenhagen}}

</ref> where it is alternatively known as the "Shang Gao theorem" (商高定理),<ref name=CY_Chen>

{{cite book |title= Early Chinese work in natural science: a re-examination of the physics of motion, acoustics, astronomy and scientific thoughts|url=https://books.google.com/?id=2Wxj0SW9hBgC&pg=PA139 |page=142 |chapter=§3.3.4 Chén Zǐ's formula and the Chóng-Chã method; Figure 40 |isbn=962-209-385-X |year=1996 |publisher=Hong Kong University Press |author1= Chen, Cheng-Yih}}

</ref> named after the Duke of Zhou's astronomer and mathematician, whose reasoning composed most of what was in the ''Zhoubi Suanjing''.<ref name=Wu>

{{cite book |title=Selected works of Wen-tsün Wu |author=Wen-tsün Wu |url=https://books.google.com/?id=xV4lECaKDzwC&pg=PA158 |page=158 |chapter=The Gougu theorem |isbn=981-279-107-8 |year=2008 |publisher=World Scientific}}

</ref>

[[Pythagoras]], whose dates are commonly given as 569–475&nbsp;BC, used algebraic methods to construct Pythagorean triples, according to [[Proclus]]'s commentary on [[Euclid]]. Proclus, however, wrote between 410 and 485&nbsp;AD. According to [[T. L. Heath|Thomas L. Heath]] (1861–1940), no specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived.<ref>{{Harv|Euclid|1956|p=351}} [https://books.google.com/?id=UhgPAAAAIAAJ&pg=PA351 page 351]</ref> However, when authors such as [[Plutarch]] and [[Cicero]] attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted.<ref name="heath144">{{harv|Heath|1921|loc=Vol I, p. 144}}: "Though this is the proposition universally associated by tradition with the name of Pythagoras, no really trustworthy evidence exists that it was actually discovered by him. The comparatively late writers who attribute it to him add the story that he sacrificed an ox to celebrate his discovery."</ref><ref name=Heath3>

An extensive discussion of the historical evidence is provided in {{Harv|Euclid|1956|p=351}} [https://books.google.com/?id=UhgPAAAAIAAJ&pg=PA351 page=351]</ref> "Whether this formula is rightly attributed to Pythagoras personally, [...] one can safely assume that it belongs to the very oldest period of Pythagorean mathematics."<ref name=Fritz/>

Around 400&nbsp;BC, according to Proclus, [[Plato]] gave a method for finding Pythagorean triples that combined algebra and geometry. Around 300&nbsp;BC, in [[Euclid's Elements|Euclid's ''Elements'']], the oldest extant [[Mathematics|axiomatic proof]] of the theorem is presented.<ref name= Aaboe>

{{cite book |title=Episodes from the early history of mathematics |author=Asger Aaboe |url=https://books.google.com/books?id=5wGzF0wPFYgC&pg=PA51 |page=51 |quote=...it is not until Euclid that we find a logical sequence of general theorems with proper proofs. |isbn=0-88385-613-1 |year=1997 |publisher=Mathematical Association of America}}

</ref>

==See also==
{{Div col|colwidth=20em}}
*[[British flag theorem]]
*[[At Dulcarnon]]
*[[Fermat's Last Theorem]]
*[[Kepler triangle]]
*[[Linear algebra]]
*[[List of triangle topics]]
*[[Lp space|L<sup>''p''</sup> space]]
*[[Nonhypotenuse number]]
*[[Parallelogram law]]
*[[Ptolemy's theorem]]
*[[Pythagorean expectation]]
*[[Pythagorean tiling]]
*[[Rational trigonometry#Pythagoras's theorem|Rational trigonometry in Pythagoras's theorem]]
{{Div col end}}

==Notes==
{{Reflist}}

==References==
{{refbegin}}
* {{Cite book |last=Bell |first=John L. |url=http://publish.uwo.ca/~jbell/ |title=The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development |publisher=Kluwer |year=1999 |isbn=0-7923-5972-0 }}
* {{Cite book |last=Euclid |title= The Thirteen Books of Euclid's Elements, Translated from the Text of Heiberg, with Introduction and Commentary |translator-last=Heath | translator-first=Thomas L. |publisher=Dover |edition=Reprint of 2nd (1925) |year= 1956 |volume= Vol. 1 (Books I and II)|ref=harv}} On-line text at [https://archive.org/details/euclid_heath_2nd_ed archive.org]
*{{cite book |title=A History of Greek Mathematics (2 Vols.)|last=Heath |first=Sir Thomas |publisher= Clarendon Press, Oxford |url=https://books.google.com/?id=h4JsAAAAMAAJ&pg=PA144 |pages=144 ''ff'' |chapter=The 'Theorem of Pythagoras' |year=1921 |edition= Dover Publications, Inc. (1981) |isbn= 0-486-24073-8|ref=harv}}
<!-- * {{Cite book |last= Kaplan|first=Robert and Ellen| title=Hidden Harmonies: The Lives and Times of the Pythagorean Theorem |location=New York, NY |publisher=Bloomsbury Press |url=http://www.bloomsburypress.com/books/catalog/hidden_harmonies_hc_220 |year=2011 |isbn=978-1-59691-522-0|ref=harv }} -->
*{{cite book |title=Euclidean and transformational geometry: a deductive inquiry |last=Libeskind |first=Shlomo |url=https://books.google.com/books?id=6YUUeO-RjU0C&pg=PA41 |isbn=0-7637-4366-6 |year=2008 |publisher=Jones & Bartlett Learning }} This high-school geometry text covers many of the topics in this WP article.
* {{Cite book |last=Loomis |first=Elisha Scott |title=The Pythagorean proposition|edition= 2nd |publisher= The National Council of Teachers of Mathematics |year=1968 |isbn=978-0-87353-036-1|ref=harv}} For full text of 2nd edition of 1940, see {{cite web|url=http://www.eric.ed.gov/PDFS/ED037335.pdf|author=Elisha Scott Loomis |title=The Pythagorean proposition: its demonstrations analyzed and classified, and bibliography of sources for data of the four kinds of proofs |accessdate=2010-05-04 |work=[[Education Resources Information Center]] |publisher=[[Institute of Education Sciences]] (IES) of the [[U.S. Department of Education]] }} Originally published in 1940 and reprinted in 1968 by National Council of Teachers of Mathematics, {{isbn|0-87353-036-5}}.
* {{Cite book |last= Maor |first=Eli |title=The Pythagorean Theorem: A 4,000-Year History |location=Princeton, New Jersey |publisher=Princeton University Press |url=https://books.google.com/?id=Z5VoBGy3AoAC&printsec=frontcover&q |year=2007 |isbn=978-0-691-12526-8|ref=harv}}
* {{cite book |title=The exact sciences in antiquity |url=https://books.google.com/?id=JVhTtVA2zr8C&pg=PA36 |author=Otto Neugebauer |isbn=0-486-22332-9 |year=1969 |edition=Republication of 1957 Brown University Press 2nd |publisher=Courier Dover Publications}}
* {{Cite book |last=Stillwell |first=John |title=Mathematics and Its History |publisher= Springer-Verlag |year=1989 |isbn=0-387-96981-0 }} Also {{isbn|3-540-96981-0}}.
*{{Cite book |last=Swetz |first= Frank |last2=Kao, T. I. |title=Was Pythagoras Chinese?: An Examination of Right Triangle Theory in Ancient China |publisher= Pennsylvania State University Press |year=1977 |isbn= 0-271-01238-2 }}
*{{Cite book |last=van der Waerden |first=Bartel Leendert |title=Geometry and Algebra in Ancient Civilizations |publisher=Springer |year=1983 |isbn=3-540-12159-5 |url=https://books.google.com/?id=_vPuAAAAMAAJ&q=%22Pythagorean+triples%22++%22Babylonian+scribes%22+inauthor:van+inauthor:der+inauthor:Waerden&dq=%22Pythagorean+triples%22++%22Babylonian+scribes%22+inauthor:van+inauthor:der+inauthor:Waerden&cd=1 |ref=harv}}
{{refend}}

==External links==
{{ProofWiki|id=Pythagoras's Theorem|title=Pythagorean theorem}}
{{Commons category}}
*{{cite book
 | last = Euclid, David E. Joyce, ed.
 | title = Elements
 | origyear = c. 300 BC
 | url = http://aleph0.clarku.edu/~djoyce/java/elements/toc.html
 | accessdate = 2006-08-30
 | year = 1997
 }} In HTML with Java-based interactive figures.
* {{springer|title=Pythagorean theorem|id=p/p075940}}
* [http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Babylonian_Pythagoras.html History topic: Pythagoras's theorem in Babylonian mathematics]
* Interactive links:
** [http://www.sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Pythagoras/pythagoras.html Interactive proof] in [[Java (programming language)|Java]] of the Pythagorean theorem
** [http://www.cut-the-knot.org/pythagoras/Perigal.shtml Another interactive proof] in [[Java (programming language)|Java]] of the Pythagorean theorem
** [http://www.mathopenref.com/pythagorastheorem.html Pythagorean theorem] with interactive animation
** [http://math.ucr.edu/~jdp/Relativity/Pythagorus.html Animated, non-algebraic, and user-paced] Pythagorean theorem
* [https://www.youtube.com/watch?v=CAkMUdeB06o Pythagorean theorem water demo] on YouTube
* [http://www.cut-the-knot.org/pythagoras/index.shtml Pythagorean theorem] (more than 70 proofs from [[cut-the-knot]])
* {{MathWorld|title=Pythagorean theorem|urlname=PythagoreanTheorem}}

{{good article}}

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