Polygon Angle Sums

Core Idea

The sum of the interior angles of an n-sided polygon is (n-2) times 180 degrees. This is derived by dividing the polygon into (n-2) triangles from one vertex. The sum of exterior angles of any convex polygon is always 360 degrees. These formulas enable finding individual angle measures, especially in regular polygons.

How It's Best Learned

Draw diagonals from one vertex of various polygons and count the triangles formed. Establish the pattern: triangle (180), quadrilateral (360), pentagon (540), etc. Derive the general formula. For exterior angles, walk around the polygon making turns and observe that you complete one full rotation (360 degrees).

Common Misconceptions

• Confusing interior and exterior angle sums.
• Using n instead of (n-2) in the formula.
• Applying the exterior angle sum of 360 to concave polygons without adjustment.
• Forgetting that exterior angle and interior angle at each vertex are supplementary.

Explainer

You already know that the angles of a triangle sum to 180°. That single fact is the engine behind everything in this topic — the polygon angle-sum formula is just a systematic way of breaking any polygon into triangles and adding up what you already know.

Pick any vertex of a polygon and draw diagonals to every other non-adjacent vertex. A quadrilateral splits into 2 triangles, a pentagon into 3, a hexagon into 4. The pattern is always (n−2) triangles for an n-sided polygon, because from one vertex you can draw n−3 diagonals, creating n−2 triangular pieces. Since each triangle contributes 180°, the total interior angle sum is (n−2) × 180°. Check: quadrilateral gives 2 × 180 = 360°, which matches what you can verify by cutting a paper quadrilateral's corners and arranging them around a point.

The exterior angle sum has an even cleaner intuition. Imagine walking along the perimeter of a convex polygon. At each vertex, you turn by the exterior angle — the supplement of the interior angle at that vertex. After completing the full loop, you've turned a total of exactly 360°, because you're back facing the same direction after one full rotation. This gives the remarkable result that the sum of exterior angles is always 360°, regardless of the number of sides. A triangle's exterior angles (120° + 120° + 120°) sum to 360°; so do a hexagon's (60° each, six of them).

These two formulas connect cleanly. The interior and exterior angles at each vertex are supplementary: interior + exterior = 180°. So the sum of all interior angles plus the sum of all exterior angles equals n × 180°. Since exterior angles sum to 360°, interior angles sum to n × 180° − 360° = (n−2) × 180° — the same formula, derived a different way. For regular polygons, every interior angle is equal, so each measures (n−2) × 180° ÷ n. For a regular hexagon: 4 × 180 ÷ 6 = 120°. This is the reason hexagonal tiles fit together perfectly at vertices — three 120° angles fill a full 360° rotation.