Solid angle

The solid angle subtended by a surface at a point is defined to be the surface area of the projection of that surface onto a unit sphere centered at that point. Solid angles so defined are called steradians and are usually denoted by $\Omega$ (Omega). Thus the solid angle of a sphere measured at its center is 4π steradians, and the solid angle subtended at the center of a cube by one of its sides is one-sixth of this, that is, 2π/3 steradians.

A solid angle is the three dimensional analogue of the ordinary angle, and the steradian is similarly analogous to the radian.

The steradian (symbol sr) is the SI unit of solid angle. Solid angles can also be measured in degrees².

One way to determine the solid angle that a surface subtends at a point is to imagine a sphere centered at the point. Now, compute the fractional area of the surface relative to the sphere by dividing the surface area of the part of the sphere that is contained within the outline of the projection of the surface onto the sphere by the total area of the sphere.

1. To obtain the solid angle in steradians or radians squared, multiply the fractional area by 4π.
2. To obtain the solid angle in degrees squared, multiply the fractional area by 4 x 180²/π which is equal to 129600/π.

By analogy with the two dimensional case —

• To get an angle, imagine two lines passing through the center of a unit circle. The length of the arc between the lines on the unit circle is the angle, in radians.

• To get a solid angle, imagine three or more planes passing through the center of a unit sphere. The area of the surface between the planes on the unit sphere is the solid angle, in steradians. The angle between two planes is termed dihedral, between three trihedral, between any number more than three polyhedral. A spherical angle is a particular dihedral angle; it is the angle between two intersecting arcs on a sphere, and is measured by the angle between the planes containing the arcs and the centre of the sphere.

Practical applications

• defining luminosity
• calculating spherical excess E of a spherical triangle
• the calculation of potentials by using the Boundary Element Method (BEM)

Solid angles for common objects

• An efficient algorithm for calculating the solid angle subtended by a triangle with vertices R1, R2 and R3, as seen from the origin has been given by Oosterom and Strackee (IEEE Trans. Biom. Eng., Vol BME-30, No 2, 1983):

$\tan \left({\frac {1}{2}}\Omega \right)={\frac {[{\mathbf {R} }_{1}{\mathbf {R} }_{2}{\mathbf {R} }_{3}]}{R_{1}R_{2}R_{3}+({\mathbf {R} }_{1}\cdot {\mathbf {R} }_{2})R_{3}+({\mathbf {R} }_{1}\cdot {\mathbf {R} }_{3})R_{2}+({\mathbf {R} }_{2}\cdot {\mathbf {R} }_{3})R_{1}}}$.

• The solid angle of a hemisphere is 2*π.

• The solid angle of a cone with apex angle a is 2*π*(1 - cos[a/2]).

• The solid angle of a four-sided right regular pyramid with apex angle a (measured to the faces of the pyramid) is 4*arccos[-(sin[a/2])^2] - 2*π.

• The sun and moon are both seen from Earth at a fractional area of 0.001% of the celestial hemisphere.