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{{short description|SI derived unit of solid angle}}
{{Infobox Unit
| name      = Steradian
| image     = [[File:Steradian.svg|150px]]
| caption   = A graphical representation of 1 steradian.{{br}}The sphere has radius ''r'', and in this case the area ''A'' of the highlighted surface patch is ''r''{{sup|2}}. The solid angle Ω equals [{{nowrap|''A''/''r''{{sup|2}}] sr}} which is {{nowrap|1 sr}} in this example. The entire sphere has a solid angle of {{gaps|4{{pi}}|sr}}.
| standard  = [[SI derived unit]]
| quantity  = [[Solid angle]]
| symbol    = sr
| units1       = SI base units
| inunits1     = 1 m<sup>2</sup>/m<sup>2</sup>
}}
The '''steradian''' (symbol: '''sr''') or '''square radian'''<ref>{{cite book |url=https://books.google.com/books?id=xhZRA1K57wIC&lpg=PA51&dq=steradian%20%22square%20radian%22&pg=PA51#v=onepage&q=steradian%20%22square%20radian%22&f=false|title=Antenna Theory and Design|isbn=978-0-470-57664-9|author1=Stutzman|first1=Warren L|last2=Thiele|first2=Gary A|date=2012-05-22}}</ref><ref>{{cite book |url=https://books.google.com/books?id=zLKQXGFUMPkC&lpg=PA11&dq=steradian%20%22square%20radian%22&pg=PA11#v=onepage&q=steradian%20%22square%20radian%22&f=false|title=Spherical Astronomy|isbn=978-0-323-14912-9|author1=Woolard|first1=Edgar|date=2012-12-02}}</ref> is the [[International System of Units|SI unit]] of [[solid angle]]. It is used in three-[[dimension]]al geometry, and is analogous to the [[radian]], which quantifies [[planar angles]]. Whereas an angle in radians, projected onto a circle, gives a ''length'' on the circumference, a solid angle in steradians, projected onto a sphere, gives an ''area'' on the surface. The name is derived from the [[Greek language|Greek]] {{lang|grc|στερεός}} {{transl|grc|stereos}} 'solid' + radian.

The steradian, like the radian, is a [[Dimensionless quantity|dimensionless]] unit, the quotient of the area subtended and the square of its distance from the center. Both the numerator and denominator of this ratio have dimension length squared (i.e. {{nowrap|1=L{{sup|2}}/L{{sup|2}} = 1}}, dimensionless). It is useful, however, to distinguish between dimensionless quantities of a different nature, so the symbol "sr" is used to indicate a solid angle. For example, [[radiant intensity]] can be measured in watts per steradian (W⋅sr<sup>−1</sup>). The steradian was formerly an [[SI supplementary unit]], but this category was abolished in 1995 and the steradian is now considered an [[SI derived unit]].

== Definition ==

A steradian can be defined as the solid angle [[subtended angle|subtended]] at the center of a [[unit sphere]] by a unit [[area]] on its surface. For a general sphere of [[radius]] ''r'', any portion of its surface with area {{nowrap|1=''A'' = ''r''<sup>2</sup>}} subtends one steradian at its center.<ref>"Steradian", ''McGraw-Hill Dictionary of Scientific and Technical Terms'', fifth edition, Sybil P. Parker, editor in chief. McGraw-Hill, 1997. {{ISBN|0-07-052433-5}}.</ref>

The solid angle is related to the area it cuts out of a sphere:
::<math>\Omega = \frac{A}{r^2}\ \mathrm{sr} \, = \frac{2\pi h}{r}\ \mathrm{sr}</math>
:where
::{{mvar|A}} is the [[surface area]] of the [[spherical cap]], <math>2\pi rh</math>,
::{{mvar|r}} is the radius of the sphere, and
::sr is the unit, steradian.

Because the surface area ''A'' of a sphere is 4{{pi}}''r''<sup>2</sup>, the definition implies that a sphere subtends 4{{pi}} steradians (≈&nbsp;12.56637&nbsp;sr) at its center. By the same argument, the maximum solid angle that can be subtended at any point is 4{{pi}}&nbsp;sr.

== Other properties ==
[[File:Steradian cone and cap.svg|thumb|right|Section of cone (1) and spherical cap (2) that subtend a solid angle of one steradian inside a sphere]]
If {{nowrap|1=''A'' = ''r''<sup>2</sup>}}, it corresponds to the area of a [[spherical cap]] ({{nowrap|1=''A'' = 2{{pi}}''rh''}}) (where ''h'' stands for the "height" of the cap) and the relationship {{nowrap|1={{sfrac|''h''|''r''}} = {{sfrac|1|2{{pi}}}}}} holds. Therefore, in this case, one steradian corresponds to the plane (i.e. radian) angle of the cross-section of a simple cone subtending the plane angle 2''θ'', with ''θ'' given by:

:<math>
\begin{align}
\theta & = \arccos \left( \frac{r-h}{r} \right)\\
       & = \arccos \left( 1 - \frac{h}{r} \right)\\
       & = \arccos \left( 1 - \frac{1}{2\pi} \right) \approx 0.572 \,\text{ rad,} \text{ or } 32.77^\circ.
\end{align}
</math>

This angle corresponds to the plane aperture angle of 2''θ'' ≈&nbsp;1.144 rad or 65.54°.

A steradian is also equal to the spherical area of a [[polygon]] having an [[angle excess]] of 1&nbsp;radian, to {{sfrac|1|4{{pi}}}} of a complete [[sphere]], or to {{resize|150%|(}}{{sfrac|180°|{{pi}}}}{{resize|150%|)}}{{su|p=2}} ≈ 3282.80635 [[square degree]]s.

The solid angle of a cone whose cross-section subtends the angle 2''θ'' is:

:<math>\Omega = 2\pi\left(1 - \cos\theta\right)\,\mathrm{sr}</math>.

== SI multiples ==
Millisteradians (msr) and microsteradians (μsr) are occasionally used to describe [[light beam|light]] and [[particle beam|particle]] beams.<ref>Stephen M. Shafroth, James Christopher Austin, ''Accelerator-based Atomic Physics: Techniques and Applications'', 1997, {{isbn|1563964848}}, p. 333</ref><ref>R. Bracewell, Govind Swarup, "The Stanford microwave spectroheliograph antenna, a microsteradian pencil beam interferometer" ''IRE Transactions on Antennas and Propagation'' '''9''':1:22-30 (1961)</ref> Other multiples are rarely used.<!--I couldn't find a single example of decasteradian in Google books and Scholar-->

== See also ==
* [[N-sphere|''n''-sphere]]
* [[Square degree]]
* [[Spat (unit)]]
* [[List of constellations by area]]

== Notes ==
{{noteslist}}

== References ==
{{Reflist}}

==External links==
{{Wiktionary}}
*{{Commonscatinline}}

{{SI units}}
{{classical mechanics derived SI units}}

[[Category:Natural units]]
[[Category:SI derived units]]
[[Category:Units of solid angle]]