Steradian

A graphical representation of 1 steradian

The steradian (symbol: sr) is the SI unit of solid angle. It is used to describe two-dimensional angular spans in three-dimensional space, analogous to the way in which the radian describes 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 because 1 sr = m2·m−2 = 1. It is useful, however, to distinguish between dimensionless quantities of different nature, so in practice the symbol "sr" is used where appropriate, rather than the derived unit "1" or no unit at all. 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.

Contents

Definition

Section of cone (1) and spherical cap (2) inside a sphere

A steradian is defined as the solid angle subtended at the center of a sphere of radius r by a portion of the surface of the sphere whose area, A, equals r2.[1]

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 solid angle of a simple cone subtending an angle θ, 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 an apex angle of 2θ ≈ 1.144 rad or 65.54°.

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.

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 the simple cone subtending an angle θ is given by:

\Omega = 2\pi(1 - \cos\frac{\theta}{2}).\quad

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.

Multiple Name Symbol
101 decasteradian dasr
100 steradian sr
10−1 decisteradian dsr
10−2 centisteradian csr
10−3 millisteradian msr
10−6 microsteradian µsr
10−9 nanosteradian nsr
10−12 picosteradian psr
10−15 femtosteradian fsr
10−18 attosteradian asr
10−21 zeptosteradian zsr
10−24 yoctosteradian ysr

References

  1. ↑ "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.
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