Polygon Interior/Exterior Angle Sums

Unit 6: Quadrilaterals and Polygons · ~15–30 min

Splitting a polygon into triangles

Every polygon can be cut into triangles by drawing diagonals from a single vertex. A triangle's angles always sum to 180°, so counting how many triangles fit inside a polygon tells you the polygon's total interior angle sum. An n-sided polygon splits into exactly n − 2 triangles this way.

• Polygon Interior Angle Sum Theorem: The sum of the interior angles of a convex polygon with n sides is (n − 2) · 180°.
• Polygon Exterior Angle Sum Theorem: The sum of the exterior angles of any convex polygon, one at each vertex, is always 360° — regardless of how many sides it has.

For a regular polygon (all sides and angles congruent), divide each sum evenly across the n vertices to get one interior or one exterior angle.

Guided practice

• 1.Find the sum of the interior angles of a nonagon (9 sides).
  Show answer

  (9 − 2) · 180° = 7 · 180° = 1260°.
• 2.A polygon has an interior angle sum of 1080°. How many sides does it have?
  Show answer

  Set (n − 2) · 180 = 1080, so n − 2 = 6, giving n = 8 sides (an octagon).
• 3.Find the measure of each exterior angle of a regular 15-gon.
  Show answer

  360° ÷ 15 = 24° per exterior angle.
• 4.Each interior angle of a regular polygon measures 156°. How many sides does the polygon have?
  Show answer

  Each exterior angle = 180° − 156° = 24°. Since exterior angles sum to 360°, n = 360° ÷ 24° = 15 sides.
• 5.Explain why the sum of a polygon's exterior angles stays 360° no matter how many sides it has, while the interior sum keeps growing.
  Show answer

  Walking around the polygon's boundary once means turning through a full 360° rotation total, one turn per vertex — that total never changes. The interior sum grows instead because each extra side adds another triangle's worth of angle, +180° per side.