A graphical representation of 1 steradian.The sphere has radius r, and in this case the area A of the highlighted surface patch is r2. The solid angle ? equals [ which is in this example. The entire sphere has a solid angle of 4?sr.
General information
Unit systemSI derived unit
Unit ofSolid angle
Symbolsr
Conversions
SI base units1 m2/m2

The steradian (symbol: sr) or square radian[1][2] is the SI unit of solid angle. It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. 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. The name is derived from the Greek ? stereos 'solid' + radian.

The steradian, like the radian, is a dimensionless unit, the quotient of 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. , 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

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 subtends one steradian at its center.[3]

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

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

Because the surface area A of a sphere is 4?r2, the definition implies that a sphere subtends 4? steradians (? 12.56637 sr) at its center. By the same argument, the maximum solid angle that can be subtended at any point is 4? sr.

## Other properties

Section of cone (1) and spherical cap (2) that subtend a solid angle of one steradian inside a sphere

If , it corresponds to the area of a spherical cap (where h stands for the "height" of the cap) and the relationship holds. Therefore, in this case, one steradian corresponds to the plane (i.e. radian) angle of the cross-section of a simple cone subtending the plane angle 2?, with ? given by:

{\displaystyle {\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 2? ? 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 of a complete sphere, or to ()2
? 3282.80635 square degrees.

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

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

## SI multiples

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

## References

1. ^ Stutzman, Warren L; Thiele, Gary A (2012-05-22). Antenna Theory and Design. ISBN 978-0-470-57664-9.
2. ^ Woolard, Edgar (2012-12-02). Spherical Astronomy. ISBN 978-0-323-14912-9.
3. ^ "Steradian", McGraw-Hill Dictionary of Scientific and Technical Terms, fifth edition, Sybil P. Parker, editor in chief. McGraw-Hill, 1997. ISBN 0-07-052433-5.
4. ^ Stephen M. Shafroth, James Christopher Austin, Accelerator-based Atomic Physics: Techniques and Applications, 1997, ISBN 1563964848, p. 333
5. ^ R. Bracewell, Govind Swarup, "The Stanford microwave spectroheliograph antenna, a microsteradian pencil beam interferometer" IRE Transactions on Antennas and Propagation 9:1:22-30 (1961)