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