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 Ω equals A sr/r2 which is 1 sr in this example. The entire sphere has a solid angle of 4πsr.
Unit information
Unit system SI derived unit
Unit of Solid angle
Symbol ㏛

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. 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 vertex: 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.[3]

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

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

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:

{\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 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:

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

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

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

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

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

So to give an example, a measurement of the width of an object seen would be given in radians. At the same time its visible area over one's visible field would be given in steradians. Just as the area of a circle is quadratically related to its diameter or radius, so is the area of the piece of a spherical surface cut out by a circular cone related to the angle across the patch. That area, or equivalently the proportion of a sphere its cone takes up, is a measure of three dimensional proportional volume, analogous to the two dimensional measure over the circle represented by an angle.

The one dimensional circular measure has the units of radians or degrees, while the two dimensional one is expressed in steradians. In higher dimensional mathematical spaces, units for analogous solid angles have not been explicitly named. When they are used, they are dealt with by analogy with the circular or spherical cases. That is, as a proportion of the relevant unit hypersphere taken up by the generalized angle, or set expressed in spherical coordinates.

SI multiples

A complete sphere subtends 4π ≈ 12.56637 steradians, so multiples larger than the decasteradian are rarely used.

Any range in excess of the whole area of a sphere would only be needed in conjunction with non-Euclidean, spherical geometry. Then it would also usually only be required in the context of nonlocal analysis. As such, solid angles over 4π are used exceedingly rarely in the literature.

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).[a]

All constellations except those of the zodiac together subtend 0.992 dasr.

100 steradian sr Area of Oceania plus Asia excluding Russia, relative to Earth (yellow on map).[b]

The Heavenly Waters constellation family subtends 1.16 sr.

10−1 decisteradian dsr Area of Algeria plus Libya, relative to Earth (green on map)[c]

The constellation Lupus subtends 1.02 dsr.

10−2 centisteradian csr Area of Zimbabwe, relative to Earth (blue on map).[d]

The smallest constellation, Crux subtends 2.09 csr.

10−3 millisteradian msr Area of Switzerland, relative to Earth (red on map).[e]

The Earth, viewed from the Moon, subtends 1.2 msr.[4]

10−6 microsteradian µsr Area of Costa Mesa, California, relative to Earth.[f]

The Sun and the Moon, viewed from Earth, each subtends 60 µsr.

Mars, viewed from Earth at its closest approach, subtends 11 nsr.[g]

10−12 picosteradian psr Area of a small apartment, relative to Earth.

Pluto, viewed from Earth at its closest approach, subtends 0.24 psr.[h]

10−15 femtosteradian fsr Area of a sheet of A5 paper, relative to Earth.

Alpha Centauri A, viewed from Earth, subtends 0.9 fsr.[i]

10−18 attosteradian asr Area of a hole made by a standard two-hole punch, relative to Earth.

Proxima Centauri, viewed from Earth, subtends 20 asr.[j]

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.

Notes

1. ^ 10.0 sr (404 million km2 out of 510 million km2)
2. ^ 1.01 sr (40.8 million km2 out of 510 million km2)
3. ^ 0.102 sr (4.14 million km2 out of 510 million km2)
4. ^ 0.00963 sr (391 000 km2 out of 510 million km2); Paraguay, at 0.0100 sr (407 000 km2) is closer to 1 csr, but has been shaded for the 10 sr region as part of the Americas
5. ^ 0.00102 sr (41 300 km2 out of 510 million km2)
6. ^ 0.00000100 sr (40.7 km2 out of 510 million km2)
7. ^ π × (25.113 / 60 / 60 / 2)2 / 3282.80635 × 1 000 000 000
8. ^ π × (0.115 / 60 / 60 / 2)2 / 3282.80635 × 1 000 000 000
9. ^ π × (0.007 / 60 / 60 / 2)2 / 3282.80635 × 1 000 000 000
10. ^ π × (0.001 / 60 / 60 / 2)2 / 3282.80635 × 1 000 000 000

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. ^ Near-side/far-side impact crater counts | NASA Lunar Science Institute