Inequalities with tetrahedral angles

A tetrahedral angle is formed at a vertex of a tetrahedron by the four faces meeting at that point. The angles between the faces (called dihedral angles) and the solid angle at the vertex are related by certain inequalities. These inequalities help us understand the possible shapes a tetrahedron can have.

Important properties

• The sum of the face angles at a vertex of a tetrahedron is less than $360^ ext{o}$.

• The sum of the dihedral angles at a vertex is greater than $180^ ext{o}$ but less than $360^ ext{o}$.

• If $A$, $B$, $C$ are the plane angles at a vertex, then $A + B + C < 360^ ext{o}$.

• The solid angle at a vertex is always less than $4\pi$ steradians (the solid angle around a point in space).