steradian

The steradian (symbolized sr) is the Standard International (SI) unit of solid angular measure. There are 4 pi, or approximately 12.5664, steradians in a complete sphere.

A steradian is defined as conical in shape, as shown in the illustration. Point P represents the center of the sphere. The solid (conical) angle q, representing one steradian, is such that the area A of the subtended portion of the sphere is equal to r², where r is the radius of the sphere.

A general sense of the steradian can be envisioned by considering a sphere whose radius is one meter (r = 1m). Imagine a cone with its apex P at the center of the sphere, and that intersects the surface in a circle (shown as a red ellipse, the upper half of which is dashed). Suppose the flare angle q of the cone is such that the area A of the spherical segment within the circle is equal to one meter squared (A = 1 m²). Then the flare angle of the cone is equal to 1 steradian (q = 1 sr). The total surface area of the sphere is, in this case, 12.5664 square meters (4 pi times the square of the radius).

Based on the foregoing example, the geometry of which is independent of scale, it can be said that a solid angle of 1 sr encompasses about 1/12.5664, or 7.9577 percent, of the space surrounding a point.

The number of steradians in a given solid angle can be determined by dividing the area on the surface of a sphere lying within the intersection of that solid angle with the surface of the sphere (when the focus of the solid angle is located at the center of the sphere) by the square of the radius of the sphere.

Also see radian and Standard International (SI) System of Units.