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The steradian (symbol: sr) or square radian is the SI unit of solid angle. It is used in three-dimensional geometry, and is analogous to the radian which quantifies planar angles. The name is derived from the Greek stereos for "solid" and the Latin radius for "ray, beam".

The steradian, like the radian, is a dimensionless unit, essentially because a solid angle is the ratio between 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. L2/L2 = 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−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 Section of cone (1) and spherical cap (2) that subtend a solid angle of one steradian inside a sphere

A steradian can be defined as the solid 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 A = r2 subtends one steradian.

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

$\Omega ={\frac {A}{r^{2}}}\mathrm {sr} \,={\frac {2\pi h}{r}}\mathrm {sr}$ where
A is the surface area of the spherical cap, $2\pi rh$ ,
r is the radius of the sphere, and

Because the surface area A of a sphere is 4πr2, the definition implies that a sphere measures 4π (≈ 12.56637) steradians. By the same argument, the maximum solid angle that can be subtended at any point is 4π sr.

## Other properties

Since A = r2, it corresponds to the area of a spherical cap (A = 2πrh) (wherein h stands for the "height" of the cap), and the relationship h/r = 1/2π holds. Therefore, one steradian corresponds to the plane (i.e. radian) angle of the cross-section of a simple cone subtending the plane angle 2θ, with θ given by:

{\begin{aligned}\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{aligned}} This angle corresponds to the plane aperture angle of 2θ ≈ 1.144 rad or 65.54°.

A steradian is also equal to the spherical area of a polygon having an angle excess of 1 radian, to 1/4π of a complete sphere, or to (180/π)2 ≈ 3282.80635 square degrees.

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

$\Omega =2\pi \left(1-\cos \theta \right)\,\mathrm {sr}$ .

In two dimensions, an angle is related to the length of the circular arc that it spans:

$\theta ={\frac {l}{r}}\,\mathrm {rad}$ where
l is arc length,
r is the radius of the circle, and

Similarly in three dimensions, the solid angle is related to the area of the spherical surface that it spans:

$\Omega ={\frac {A}{r^{2}}}\,\mathrm {sr}$ where
A is the surface area of the spherical cap, 2πrh,
r is the radius of the sphere, and