solid angle

A 3-D analogue to an ordinary angle. It represents the portion of the space around a vertex of a cone, polyhedron, or other solid figure which is enclosed by the vertex. To be specific, it is the area of the portion of the surface of a unit sphere centered at the vertex which is enclosed by the extensions of the surfaces meeting at the vertex.

The solid angle is the extension of the concept of angle to three dimensions.

Analogous to the way 'normal' (2D) angles (planar angles) are associated with pie-shaped slices (sectors) of a circle, solid angles are associated with cone-shaped slices of a sphere ¹. Solid angles are measured in steradians (abbreviated as sr). There are 4π steradians in a sphere, just as there are 2π radians in a circle.

So how does one measure solid angles? To answer this question it is instructive to reconsider the familiar 2D case. In the case of the planar angle, there are two lines meeting at a vertex at a certain angle. Here is one way to measure that angle:

1. Draw a circle around that vertex -- any radius will do.
2. Figure out where the circle intersects the two lines.
3. Measure the length of the arc between the two intersections.
4. Divide the length of the arc by the radius to get the angle in radians.

So we have that

θ = s / r

where θ is the angle, s is the arc length, and r is the radius. Since the circumference of a circle is 2πr, there are 2π radians in a circle.

For the 3D case, the situation is similar. Instead of lines meeting at a vertex, we have surfaces meeting at a vertex. To measure the solid angle, then we:

1. Draw a sphere around that vertex -- any radius will do.
2. Figure out where the sphere intersects the surfaces.
3. Measure the area of the patch on the sphere bounded by the intersections.
4. Divide the area by the square of the radius to get the solid angle in steradians.

So we have that

ω = A / r ²

where ω is the solid angle, A is the area, and r is the radius. The surface area of a sphere is 4πr ². Therefore, there are 4π steradians in a sphere.

Now for some examples of solid angles:

• Interior solid angle at corner of a cube: 4π/8 = π/2 sr
• Interior solid angle at corner two flat surfaces meeting at a right angle: 4π/4 = π sr
• Solid angle at any point on a plane: 4π/2 = 2π sr

See also: differential solid angle.

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¹ The cone need not have a circular cross section.