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Steradian

SI derived unit of solid angle

steradian
A graphical representation of two different steradians. The sphere has radius r, and in this case the area A of the highlighted spherical capisr2. The solid angle Ω equals [A/r2] sr which is 1 sr in this example. The entire sphere has a solid angle of 4π sr.
General information
Unit system	SI
Unit of	solid angle
Symbol	sr
Conversions
1 srin ...	... is equal to ...
SI base units	1 m2/m2
square degrees	⁠1802/π2⁠ deg2 ≈ 3282.8 deg2

The steradian (symbol: sr) or square radian^[1][2] 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 angles. A solid angle in steradians, projected onto a sphere, gives the area of a spherical cap on the surface, whereas an angle in radians, projected onto a circle, gives a length of a circular arc on the circumference. The name is derived from the Greek στερεός stereos 'solid' + radian.

The steradian is a dimensionless unit, the quotient of the area subtended and the square of its distance from the centre. Both the numerator and denominator of this ratio have dimension length squared (i.e. L²/L² = 1, dimensionless). It is useful, however, to distinguish between dimensionless quantities of a different kind, such as the radian (a ratio of quantities of dimension length), so the symbol "sr" is used to indicate a solid angle. For example, radiant intensity can be measured in watts per steradian (W⋅sr⁻¹). 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[edit]

A steradian can be defined as the solid angle subtended at the centre 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 A = r² subtends one steradian at its centre.^[3]

The solid angle is related to the area it cuts out of a sphere:

${\displaystyle \Omega ={\frac {A}{r^{2}}}\ {\text{sr}}\,={\frac {2\pi h}{r}}\ {\text{sr}},}$

where

• Ω is the solid angle
• A is the surface area of the spherical cap, ${\displaystyle 2\pi rh}$,
• r is the radius of the sphere,
• h is the height of the cap, and
• sr is the unit, steradian.

Because the surface area A of a sphere is 4πr², the definition implies that a sphere subtends 4π steradians (≈ 12.56637 sr) at its centre, or that a steradian subtends 1/4π ≈ 0.07958 of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is 4πsr.

Other properties[edit]

The area of a spherical capisA = 2πrh, where h is the "height" of the cap. If A = r², then ${\displaystyle {\tfrac {h}{r}}={\tfrac {1}{2\pi }}}$. From this, one can compute the plane aperture angle 2θ of the cross-section of a simple cone whose solid angle equals one steradian:

${\displaystyle \theta =\arccos \left({\frac {r-h}{r}}\right)=\arccos \left(1-{\frac {h}{r}}\right)=\arccos \left(1-{\frac {1}{2\pi }}\right),}$

giving θ ≈ 0.572 rad or 32.77° and 2θ ≈ 1.144 rad or 65.54°.

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

${\displaystyle \Omega =2\pi (1-\cos \theta ){\text{ sr}}=4\pi \sin ^{2}\left({\frac {\theta }{2}}\right){\text{ sr}}.}$

A steradian is also equal to ${\displaystyle {\tfrac {1}{4\pi }}}$ of a complete sphere (spat), to ${\displaystyle \left({\tfrac {360^{\circ }}{2\pi }}\right)^{2}}$ ≈ 3282.80635 square degrees, and to the spherical area of a polygon having an angle excess of 1 radian.^{[clarification needed]}

SI multiples[edit]

Millisteradians (msr) and microsteradians (μsr) are occasionally used to describe light and particle beams.^[4][5] Other multiples are rarely used.

See also[edit]

• n-sphere
• Spat (angular unit)
• IAU designated constellations by area

References[edit]

External links[edit]

Look up steradian in Wiktionary, the free dictionary.

• Media related to Steradian at Wikimedia Commons