Steradian

Steradian
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 squared radian is the SI unit of solid angle. It is used in three-dimensional space, and functions analogously to the manner in which the radian 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 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 length squared (i.e., L²/L² = Φ - no unit). 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) 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.^[1]

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

\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

sr is the unit, steradian.

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

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

Analogue to radians

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

\theta = \frac{l}{r}\,\mathrm{rad}

where

l is arc length,

r is the radius of the circle, and

rad is the radian.

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

\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

sr is the steradian.

SI multiples

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

Solid angle of various areas relative to Earth

See also

• List of constellations by area
• Square degree

Notes and references

SI units Base units ampere candela kelvin kilogram metre mole second Derived units with special names becquerel coulomb degree Celsius farad gray henry hertz joule katal lumen lux newton ohm pascal radian siemens sievert steradian tesla volt watt weber Other accepted units astronomical unit dalton day decibel degree Celsius degree of arc electronvolt hectare hour litre minute minute of arc neper second of arc tonne atomic units natural units See also Conversion of units History of the metric system Metric prefixes Proposed redefinitions Systems of measurement Book Category Outline