Angles in Any Polygon

Part of the Polygons & Quadrilaterals section of Coddy's Geometry journey. Lesson 3 of 61.

The diagonals from one vertex cut any convex polygon (one with no dent) into triangles. A vertex has no diagonal to itself or its two neighbours, so a polygon with n sides is cut into n − 2 triangles: a pentagon into 3, a hexagon into 4.

$Pentagon ABCDE cut into three triangles by the diagonals from A$

Each triangle brings 180°, so the angles of a polygon with n sides add up to (n - 2) × 180 degrees. A quadrilateral: 360°. A pentagon: 3 × 180 = 540. A hexagon: 4 × 180 = 720. A polygon with a dent still has the same total: it can be cut into n − 2 triangles too, just not all from one corner.

A pentagon with angles 100°, 120°, 95° and 115° has a fifth angle of 540 - 430 = 110 degrees. On the board, tap the unknown angle and pick Angles in a polygon. The board counts the sides for you.