View source for Steradian

{{short description|SI derived unit of solid angle}}
{{Infobox Unit
| name      = steradian
| image     = [[File:Solid_Angle,_1_Steradian.svg|150px]]
| caption   = A graphical representation of two different steradians.{{br}}The sphere has radius {{math|''r''}}, and in this case the area {{math|''A''}} of the highlighted [[spherical cap]] is {{math|''r''{{sup|2}}}}. The solid angle {{math|Ω}} equals {{math|[''A''/''r''{{sup|2}}] sr}} which is {{math|1 sr}} in this example. The entire sphere has a solid angle of {{math|4''π'' sr}}.
| standard  = [[SI]]
| quantity  = [[solid angle]]
| symbol    = sr
| units1    = SI base units
| inunits1  = 1 m<sup>2</sup>/m<sup>2</sup>
| units2    = [[square degree]]s
| inunits2  = {{sfrac|{{val|180}}<sup>2</sup>|{{pi}}<sup>2</sup>}}&nbsp;<span>deg<sup>2</sup></span>{{br}}&asymp; {{val|3282.8}}&nbsp;deg<sup>2</sup>
}}

The '''steradian''' (symbol: '''sr''') or '''square radian'''<ref>{{cite book |url=https://books.google.com/books?id=xhZRA1K57wIC&q=steradian%20%22square%20radian%22&pg=PA51 |title=Antenna Theory and Design |isbn=978-0-470-57664-9 |last1=Stutzman |first1=Warren L |last2=Thiele |first2=Gary A |date=2012-05-22 |publisher=John Wiley & Sons }}</ref><ref>{{cite book |url=https://books.google.com/books?id=zLKQXGFUMPkC&q=steradian%20%22square%20radian%22&pg=PA11 |title=Spherical Astronomy |isbn=978-0-323-14912-9 |last1=Woolard |first1=Edgar |date=2012-12-02|publisher=Elsevier }}</ref> is the unit of [[solid angle]] in the [[International System of Units]] (SI). It is used in [[three-dimensional geometry]], and is analogous to the [[radian]], which quantifies [[planar angle]]s. A solid angle in the form of a [[circular cone]] can be projected onto a sphere from its centre, delineating a [[spherical cap]] where the cone intersects the sphere. The magnitude of the solid angle expressed in steradians is defined as the quotient of the surface area of the spherical cap and the square of the sphere's radius. This is analogous to the way a plane angle projected onto a circle delineates a [[circular arc]] on the circumference, whose length is proportional to the angle. Steradians can be used to measure a solid angle of any projected shape. The solid angle subtended is the same as that of a cone with the same projected area. A solid angle of one steradian subtends a [[cone aperture]] of approximately 1.144 radians or 65.54 degrees.

In the SI, solid angle is considered to be a [[Dimensionless quantity|dimensionless]] quantity, the ratio of the area projected onto a surrounding sphere and the square of the sphere's radius. This is the number of square radians in the solid angle. This means that the SI steradian is the number of square radians in a solid angle equal to one square radian, which of course is the number one. It is useful to distinguish between dimensionless quantities of a different [[Kind of quantity|kind]], such as the radian (in the SI, a ratio of quantities of dimension length), so the symbol sr is used.  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]].

The name ''steradian'' is derived from the [[Greek language|Greek]] {{lang|grc|στερεός}} {{Transliteration|grc|stereos}} 'solid' + radian.

[[File:BlankMap-World6 steradian.svg|thumb|Solid angle of countries and other entities relative to the centre of Earth.]]

== Definition ==

A steradian can be defined as the solid angle [[subtended angle|subtended]] at the centre of a [[unit sphere]] by a unit [[area]] (of any shape) on its surface. For a general sphere of [[radius]] {{math|''r''}}, any portion of its surface with area {{math|1=''A'' = ''r''<sup>2</sup>}} subtends one steradian at its centre.<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>

A solid angle in the form of a circular cone is related to the area it cuts out of a sphere:
: <math>\Omega = \frac{A}{r^2}\ \text{sr} \, = \frac{2\pi h}{r}\ \text{sr},</math>
where
* {{math|Ω}} is the solid angle
* {{mvar|A}} is the [[surface area]] of the [[spherical cap]], {{math|2''πrh''}},
* {{mvar|r}} is the radius of the sphere,
* {{mvar|h}} is the height of the cap, and
* sr is the unit, steradian; sr = rad{{sup|2}}.

Because the surface area {{math|''A''}} of a sphere is {{math|4''πr''<sup>2</sup>}}, the definition implies that a sphere subtends {{math|4''π''}} steradians (≈&nbsp;12.56637&nbsp;sr) at its centre, or that a steradian subtends {{math|1/4''π''}}&nbsp;≈&nbsp;0.07958 of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is {{math|4''π''}}&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]]

The area of a [[spherical cap]] is {{math|1=''A'' = 2''πrh''}}, where {{mvar|h}} is the "height" of the cap. If {{math|1=''A'' = ''r''<sup>2</sup>}}, then {{math|1={{sfrac|''h''|''r''}} = {{sfrac|1|2''π''}}}}. From this, one can compute the [[cone aperture]] (a plane angle) {{math|2''θ''}} of the cross-section of a simple [[Spherical sector|spherical cone]] whose solid angle equals one steradian:
: <math>\theta = \arccos \left( \frac{r-h}{r} \right) = \arccos \left( 1 - \frac{h}{r} \right) = \arccos \left( 1 - \frac{1}{2\pi} \right) ,</math>
giving {{math|''θ''}} ≈ 0.572&nbsp;rad {{=}} 32.77° and aperture {{math|2''θ''}} ≈ 1.144&nbsp;rad {{=}} 65.54°.

The solid angle of a spherical cone whose cross-section subtends the angle {{math|2''θ''}} is:
: <math>\Omega = 2\pi(1 - \cos\theta) \text{ sr} = 4\pi\sin^2\left(\frac{\theta}{2}\right) \text{ sr}.</math>

A steradian is also equal to {{math|{{sfrac|1|4''π''}}}} of a complete [[sphere]] ([[Spat (angular unit)|spat]]), to {{math|{{resize|160%|(}}{{sfrac|360°|2''π''}}{{resize|160%|)}}{{su|p=2}}}} ≈ 3282.80635&nbsp;[[square degree]]s, and to the spherical area of a [[polygon]] having an [[angle excess]] of 1&nbsp;radian.{{clarify|date=June 2024}}

== SI multiples ==
Millisteradians (msr) and microsteradians (μsr) are occasionally used to describe [[light beam|light]] and [[particle beam|particle]] beams.<ref>{{citation |author1=Stephen M. Shafroth |author2=James Christopher Austin |title=Accelerator-based Atomic Physics: Techniques and Applications |date=1997 |isbn=1563964848 |page=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]]
* [[Spat (angular unit)]]
* [[IAU designated constellations by area]]

== References ==
{{reflist}}

== External links ==
{{Wiktionary}}
* {{Commons category-inline}}

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

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