Polygon Interior & Exterior Angle Formulas

(n − 2) × 180° for any polygon, 360° always for exterior

Reviewed by [email protected], Geometry Calculator Developer & Online Math Educator Last updated May 8, 2026

Every convex polygon's angles follow predictable formulas based on the number of sides n. Two facts to remember: the interior angles always sum to (n − 2) × 180°, and the exterior angles always sum to exactly 360° regardless of n.

The Formulas

Name Formula Notes
Interior Angle Sum S = (n − 2) × 180° For ANY polygon. n = number of sides.
Each Interior Angle (regular) a = (n − 2) × 180° / n Only for regular polygons (all sides + angles equal).
Exterior Angle Sum 360° (always) Independent of n. Always 360° for any convex polygon.
Each Exterior Angle (regular) e = 360° / n Hexagon → 60°, octagon → 45°.
Interior + Exterior Pair a + e = 180° They're supplementary at every vertex.
Number of Sides from S n = S / 180° + 2 Inverse — given the angle sum, recover n.

Worked Examples

Example 1: Hexagon (n = 6)

  1. Interior sum = (6 − 2) × 180° = 4 × 180° = 720°
  2. Each interior (regular) = 720° / 6 = 120°
  3. Each exterior = 360° / 6 = 60°
  4. Check: 120° + 60° = 180° ✓

Example 2: Find n if interior sum is 1440°

  1. n = S/180° + 2 = 1440°/180° + 2
  2. n = 8 + 2 = 10 sides (decagon)

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