Steradian

A graphical representation of 1 steradian. The sphere has radius r, and in this case the area of the patch on the surface is A = r². The solid angle is θ = A/r² so in this case θ = 1. The entire sphere has a solid angle of 4π sr ≈ 12.56637 sr.

The steradian (symbol: sr) is the SI unit of solid angle. It is used to quantify two-dimensional angular spans in three-dimensional space, analogously to how the radian quantifies angles in a plane. The name is derived from the Greek stereos for "solid" and the Latin radius for "ray, beam".

The steradian, like the radian, is dimensionless, essentially because a solid angle is the ratio between the area subtended and the square of its distance from the vertex: both the numerator and denominator of this ratio have dimension length². It is useful, however, to distinguish between dimensionless quantities of different nature, so in practice 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 from the SI in 1995 and the steradian is now considered an SI derived unit.

Definition

Section of cone (1) and spherical cap (2) 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.

Because the surface area 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. A steradian can also be called a squared radian.

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:

$\begin{align} \theta & = \arccos \left( \frac{r-h}{r} \right)\\ & = \arccos \left( 1 - \frac{h}{r} \right)\\ & = \arccos \left( 1 - \frac{1}{2\pi} \right) \approx 0.572 \,\text{ rad,} \mbox{ or } 32.77^\circ. \end{align}$

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 in steradians of a cone whose cross-section subtends the angle 2θ (θ shown in the image) is:

$\Omega = 2\pi\left(1 - \cos\left(\theta\right)\right).$

This can be re-expressed as the πR^2 circular area of the base of the cone created by θ:

$\Omega = \pi\left[\sin\left(\theta\right)\right]^2$.

Analogue to radians

In two dimensions, the angle in radians is related to the arc length it cuts out:

$\theta = \frac{l}{r} \,$

where

l is arc length, and

r is the radius of the circle.

Now in three dimensions, the solid angle in steradians is related to the area it cuts out:

$\Omega = \frac{S}{r^2} \,$

where

S is the surface area, and

r is the radius of the sphere.

SI multiples

Steradians only go up to 4π ≈ 12.56637, so the large multiples are not usable for the base unit, but could show up in such things as rate of coverage of solid angle, for example.

Solid angle of various areas relative to Earth

Multiple	Name	Symbol	May be visualized as...
101	decasteradian	dasr	Surface area of the Americas plus liquid water on Earth, relative to Earth (cyan on map)
100	steradian	sr	Area of Oceania plus Asia excluding Russia, relative to Earth (yellow on map)
10−1	decisteradian	dsr	Area of Algeria plus Libya, relative to Earth (green on map)
10−2	centisteradian	csr	Area of Zimbabwe, relative to Earth (blue on map)
10−3	millisteradian	msr	Area of Switzerland, relative to Earth (red on map)
10−6	microsteradian	µsr	Area of Costa Mesa, California, relative to Earth
10−9	nanosteradian	nsr	About 8 American football fields, relative to Earth
10−12	picosteradian	psr	Area of a small apartment, relative to Earth
10−15	femtosteradian	fsr	Area of a sheet of A5 paper, relative to Earth
10−18	attosteradian	asr	Area of a quarter-inch square, relative to Earth
10−21	zeptosteradian	zsr	Cross-sectional area of 32 gauge wire, relative to Earth
10−24	yoctosteradian	ysr	Surface area of a red blood cell, relative to Earth