# Supplementary Solid Angles for Trihedron

Supplementary Solid Angles for Trihedron

This Demonstration constructs a supplementary solid angle for a given trihedral solid angle. Let , and be the edges of a trihedron that determines the solid angle. The plane angles opposite the edges are denoted , , and the dihedral angles at the edges are denoted , , . Let be a point inside the trihedron and denote its orthogonal projections onto the faces of the trihedron by , and . Then , and are edges of a trihedron that determines the supplementary space angle.

OA

OB

OC

a

b

c

A

B

C

O'

A'

B'

C'

O'A'

O'B'

O'C'

The plane angles of the supplementary angle are , and , and its dihedral angles are , and .

π-A

π-B

π-C

π-a

π-b

π-c

The measure of the initial trihedral angle is (the spherical excess formula for a trihedron), while the measure of its supplementary angle is .

A+B+C-π

2π-(A+B+C)=(π-A)+(π-B)+(π-C)-π