A whole sphere contains about 12.57 steradians, the unit of solid angle
A radian measures a slice of a circle; a steradian measures a cone's worth of a sphere. Stand at the centre of any sphere and the whole surrounding view spans four pi steradians, about 12.57. One steradian corresponds to a cone opening of roughly 65.5 degrees, about 8 percent of everything around you.
The name joins Greek stereos, solid, with radian. Picture a cone with its tip at a sphere's centre. Where it meets the sphere it cuts out a cap, and the solid angle in steradians is that cap's area divided by the radius squared. Put simply, on a sphere of radius r, any patch of area r squared, whatever its shape, subtends one steradian. It works for shapes other than cones too: any projected outline has the same solid angle as a cone covering the same area.
Some tidy numbers fall out. Since a sphere's surface is four pi times the radius squared, the full sphere spans four pi steradians, the largest solid angle possible from any point, and a single steradian covers about 0.0796 of it. For a simple cone to enclose exactly one steradian, its half-angle works out to about 0.572 radians, or 32.77 degrees, giving the full opening of about 1.144 radians. A steradian also equals roughly 3,283 square degrees.
In the SI, solid angle is a dimensionless ratio, area over area, so a steradian is technically just the number one, counted in square radians. The symbol sr is kept to avoid confusing it with other dimensionless quantities such as the radian, and it shows up in units like watts per steradian for radiant intensity. It was once classed as an SI supplementary unit, but that category was abolished in 1995 and the steradian became a derived unit.
Physicists describing narrow beams of light or particles sometimes reach for millisteradians and microsteradians, though other multiples are rarely seen. Another view of the same unit: it is the area of a spherical polygon whose angles exceed a flat polygon's by one radian.
Source: Steradian
