- Steradian
The

**steradian**(symbol:**sr**) is theSI unit ofsolid angle . It is used to describe two-dimensional angular spans in three-dimension al space, analogous to the way in which theradian 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 = m

^{2}·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. As an example,radiant intensity can be measured in watts per steradian (W·sr^{-1}).**Definition**A single unit of steradian is defined as the

solid angle subtended at the center of asphere ofradius "r" by a portion of the surface of the sphere having anarea r^{2}.If this area, A, is equal to r

^{2}and it corresponds to the area of aspherical cap (A = 2πrh,) then the relationship $frac\{h\}\{r\}=frac\{1\}\{2pi\}$ holds. Then the solid angle of the simple cone subtending an angle θ is equal to::$egin\{align\}\; heta\; =\; arccos\; left(\; frac\{r-h\}\{r\}\; ight)\backslash \; =\; arccos\; left(\; 1\; -\; frac\{h\}\{r\}\; ight)\backslash \; =\; arccos\; left(\; 1\; -\; frac\{1\}\{2pi\}\; ight)\; approx\; 0.572\; ,\; ext\{rad\}\; mbox\{\; or\; \}\; 32.77^circend\{align\}$

This angle corresponds to an apex angle of 2θ ≈ 1.144 rad or 65.54°.

Because the surface area of this sphere is 4πr

^{2}, the definition implies that a sphere measures 4π steradians. By the same argument, the maximum solid angle that can be subtended at any point is 4πsr. A steradian can also be called a**squared radian**.A steradian is also equal to the spherical area of a

polygon having anangle excess of 1 radian, to 1/(4π) of a completesphere , or to (180/π)² or 3282.80635square degree s.The steradian was formerly an

SI supplementary unit , but this category was abolished from theSI in 1995 and the steradian is now considered anSI derived unit .**Analogue to radians**In two dimensions, the angle in radians is related to the

arc length it cuts out:::$heta\; =\; frac\{s\}\{r\}\; ,$:where::"s" 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.

**I multiples****See also***

Solid angle

*Wikimedia Foundation.
2010.*