semiexterior

Semiexterior is a term encountered in some discussions of polygon geometry, referring to an angle associated with a vertex formed by the extension of one of the sides that meet there. In many sources, the semiexterior angle at a vertex is taken to be the exterior angle that is supplementary to the interior angle at that vertex; equivalently, its measure is 180 degrees minus the interior angle.

Because the semiexterior angle is defined as the linear-pair supplement of the interior angle, it follows that at each vertex i with interior angle Ai, the semiexterior angle Si satisfies Si = 180° − Ai. For a polygon with n vertices, the sum of the interior angles is (n − 2) × 180°, and the sum of the semiexterior angles is n × 180° − (n − 2) × 180° = 360°. In triangles, the semiexterior angle at a vertex coincides with the standard exterior angle.

Examples help illustrate the idea. In an equilateral triangle, each interior angle is 60°, so each semiexterior angle is 120°, and the sum of the three semiexterior angles is 360°. In a square, each interior angle is 90°, so each semiexterior angle is 90°, and again their sum is 360°.

Usage notes: the term semiexterior is not universally standardized. Some authors reserve it for a slightly different construction, while others use it as a synonym for an exterior angle. When encountering the term, it is important to confirm the specific definition being used. See also interior angle and exterior angle.