Polygon angles

Why do the exterior angles add up to 360°? At each of the n vertices, interior + exterior = 180°. All together: n × 180° = (interior sum) + (exterior sum). The interior sum is (n − 2) × 180°, so the exterior sum is n × 180° − (n − 2) × 180° = 2 × 180° = 360°. This holds for every convex polygon, regular or not: walk once around the outside, and you turn through a full circle. For irregular polygons, subtract the known exterior angles from 360°. The number of diagonals of a convex polygon is n(n − 3) ÷ 2: from each vertex you can draw n − 3 diagonals (not to itself or its two neighbours), and each diagonal is counted twice. A hexagon has 6 × 3 ÷ 2 = 9. If a polygon has 9 diagonals, then n(n − 3) = 18, n² − 3n − 18 = 0, (n − 6)(n + 3) = 0, so n = 6 (n = −3 is impossible).