QUESTION #174

# How is the 'STERADIAN' defined and how is it used?

Asked by: Undisclosed Visitor

### Answer

Steradians are a measure of the angular 'area' subtended by a two dimensional surface about the origin in three dimensional space, just as a radian is a measure of the angle subtended by a one dimensional line about the origin in two dimensional (plane) space. Steradians are equivalently referred to as 'square radians.'

A sphere subtends 4 pi square radians (steradians) about the origin. By analogy, a circle subtends 2 pi radians about the origin. Numerically, the number of steradians in a sphere is equal to the surface area of a sphere of unit radius. I.e., area of sphere = 4 pi r^2, but with r = 1, area = 4 pi. Likewise, numerically, the number of radians in a circle is equal to the circumference of a circle of unit radius. I.e., circumference = 2 pi r, but with r = 1, circumference = 2 pi.

As one would expect, steradians (square radians) can be converted to square degrees by multiplying by the square of the number of degrees in a radian = 57.2957795... degrees. For example, the number of square degrees in a sphere is equal to 4 pi x (57.2957795)^2 = 41,253 square degrees (rounded to the nearest square degree). For those who prefer to work in square degrees, it is helpful to remember that the number of square degrees in a sphere contains the digits 1 through 5, with no repeats.

Steradians occur virtually anywhere in physics where a flux through a three dimensional surface is involved. For example, the ubiquitous factors of 4 pi that keep popping up in formulas derived in electromagnetics really just represent the scaling, or normalizing, of whatever is being described to the angular area subtended by a sphere. Not surprisingly, steradians find heavy use in antenna engineering to characterize such properties as the 'directivity' of an antenna relative to an 'isotropic' radiator (one that radiates uniformly in all directions through the surface of an imaginary sphere).
Answered by: Warren Davis, Ph.D., President, Davis Associates, Inc., Newton, MA USA