dihedralangle

The dihedral angle is the angle between two planes. It is defined along their line of intersection: in each plane take a line that is perpendicular to the intersection line, and measure the angle between these two lines.

If n1 and n2 are unit normal vectors to the two planes, the dihedral angle φ can be computed as φ = arccos( (n1 · n2) / (|n1||n2|) ). This yields φ in the range from 0 to π. In some contexts, the acute angle between planes is used, given by φ′ = arccos( |n1 · n2| / (|n1||n2|) ), which lies in [0, π/2].

In the context of polyhedra, the dihedral angle at an edge is the angle between the two faces that share that edge; the interior dihedral angle is measured inside the solid and lies between 0 and π. For example, a cube has dihedral angles of π/2 (90 degrees), while a regular tetrahedron has a dihedral angle of arccos(1/3) ≈ 70.5 degrees.

In chemistry and molecular modeling, the term dihedral angle (often called the torsion angle) describes the angle between the planes formed by four consecutive atoms (A–B–C–D). This angle governs conformations and steric interactions in molecules and is a key parameter in potential energy surfaces and rotational barriers.

The word dihedral comes from di- meaning two and hedra meaning faces or planes, reflecting its geometric origin. See also torsion angle and the general concept of angle between planes.