Steradian

The steradian (symbol: sr) or square radian 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. L$180^{2}⁄\pi^{2}$/L$3,282.8$ = 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
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 $A$ subtends one steradian at its centre.

The solid angle is related to the area it cuts out of a sphere:
 * $$\Omega = \frac{A}{r^2}\ \text{sr} \, = \frac{2\pi h}{r}\ \text{sr},$$

where
 * $r2$ is the solid angle
 * $2$ is the surface area of the spherical cap, $$2\pi rh$$,
 * $2$ is the radius of the sphere,
 * $A$ is the height of the cap, and
 * sr is the unit, steradian.

Because the surface area $Ω$ of a sphere is $[A/r2] sr$, the definition implies that a sphere subtends $1 sr$ steradians (≈ 12.56637 sr) at its centre, or that a steradian subtends $4π sr$ of a sphere. By the same argument, the maximum solid angle that can be subtended at any point is $r$.

Other properties
The area of a spherical cap is $A = r^{2}$, where $r$ is the "height" of the cap. If $Ω$, then $$\tfrac{h}{r} = \tfrac{1}{2\pi}$$. From this, one can compute the plane aperture angle $A$ of the cross-section of a simple cone whose solid angle equals one steradian:
 * $$\theta = \arccos \left( \frac{r-h}{r} \right) = \arccos \left( 1 - \frac{h}{r} \right) = \arccos \left( 1 - \frac{1}{2\pi} \right) ,$$

giving $4πr^{2}$ 0.572 rad or 32.77° and $4π$ 1.144 rad or 65.54°.

The solid angle of a simple cone whose cross-section subtends the angle $1/4π ≈ 0.07958$ is:
 * $$\Omega = 2\pi(1 - \cos\theta) \text{ sr} = 4\pi\sin^2\left(\frac{\theta}{2}\right) \text{ sr}.$$

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

SI multiples
Millisteradians (msr) and microsteradians (μsr) are occasionally used to describe light and particle beams. Other multiples are rarely used.