Interior angles

Interior angles are the angles formed inside a polygon where two sides meet. A triangle has three interior angles, a quadrilateral has four, and in general a polygon with n sides has n interior angles. The term also describes the angles that lie between two lines cut by a transversal, but on standardized tests it most often refers to polygon angles.

The essential formula is the polygon interior angle sum theorem: the interior angles of an n-sided polygon add up to (n − 2) × 180°. It works because any polygon can be split into n − 2 triangles by drawing diagonals from one vertex, and each triangle contributes 180°. So a triangle's angles sum to 180°, a quadrilateral's to 360°, a pentagon's to 540°, and a hexagon's to 720°.

For a regular polygon — all sides and angles equal — each interior angle measures (n − 2) × 180° / n. A regular hexagon's interior angles are each 720° / 6 = 120°, and a square's are each 360° / 4 = 90°. A related fact worth memorizing: the exterior angles of any polygon always sum to 360°, and each interior angle plus its exterior angle equals 180°.

Interior angle problems are a staple of standardized test geometry. The GRE, SAT, and CLT all ask questions built on the angle sum formula — finding a missing angle, computing the angles of a regular polygon, or working backward from an angle sum to the number of sides. Knowing (n − 2) × 180° cold turns these into quick arithmetic.