WOLFRAM NOTEBOOK

Solid and Dihedral Angles of a Tetrahedron

↓

dihedral

edge lengths

edge

∠ / π

1/2

1/3

1/3

1/3

1/3

1/2

2 × angle sum:

2∑d

14/3

vertex

solid

coordinates

vertex

∠ / π

-1,-0.35,-0.35

1/6

1,-0.35,-0.35

1/6

0,1.06,-0.35

1/6

0,-0.35,1.06

1/6

solid angle sum

∑s

2/3

2∑d - ∑s

use special tetra

During an eclipse, the Moon and Sun appear to have roughly the same size viewed from Earth. The field of view of an object is called the solid angle. A unit sphere around the observer has a solid angle of

4π

steradians, the same as the surface area.

Let

ABCD

be a tetrahedron and consider the solid angles defined by an observer at each vertex looking at its opposite triangle. You might think that the smallest solid angle corresponds to the smallest of the four triangles, but that is not necessarily so—it depends on the inclination of the triangle relative to the observer at the vertex.

When two planes intersect, the angle between them is called the dihedral angle. In a cube, the dihedral angles are

π/4

(or 90°). The maximum possible dihedral angle is

2π

For a triangle in the plane, let

be an interior angle. Then

2Σa=2π

, the full circle.

For a tetrahedron in 3D space, let

be a dihedral angle and

be a solid angle. Then

2Σd-Σs=4π