steradian

SI coherent derived unit with special name and symbol

Name	Symbol	Derived quantity	Expressed in terms of SI base units
steradian	sr	solid angle	m2/m2
Definition The steradian, symbol sr, is the SI coherent derived unit for solid angle. One steradian is defined as the solid angle subtended at the centre of a unit sphere by a unit area of its surface. For a sphere of radius r, any portion of its surface with area A = r2 subtends one steradian at its centre.

For a unit sphere, with a radius of one metre, a solid angle of one steradian at the centre of the sphere encloses an area of one square metre on the surface.

The magnitude in steradians of a solid angle Ω subtended at the centre of a sphere is equal to the ratio of the area of the surface A enclosed by the solid angle to the square of the length of the sphere’s radius r.

Conversely, the area A of the surface of a sphere enclosed by a solid angle at the sphere’s centre is equal to the product of the square of the radius r of the sphere and the magnitude in steradians Ω of the solid angle.

It follows that the magnitude in steradians of a complete sphere is equal to the area of the surface, 4πr², divided by the square of the length of the radius r.

The steradian is used to quantify solid angles in three-dimensional geometry. It is analogous to the radian, which quantifies planar angles in two-dimensional space. 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.

Examples of solid angles

spheres	steradians in terms of τ (τ = 2π)	steradians in terms of π
1⁄4	τ⁄2	π
1⁄3	2τ⁄3	4π⁄3
1⁄2	τ	2π
3⁄4	3τ⁄2	3π
1	2τ	4π

Usage

The steradian corresponds to the ratio of two squared lengths. However, the steradian must only be used to express solid angles, and not to express ratios of squared lengths in general.