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Angles in polygons
Any polygon can be cut into triangles. That one idea gives you every angle fact you need.
Interior angle sum
From one corner, a polygon with n sides splits into (n − 2) triangles.
Each triangle has 180°, so the interior angles add up to (n − 2) × 180°.
Pentagon: (5 − 2) × 180° = 540°
Each triangle has 180°, so the interior angles add up to (n − 2) × 180°.
Pentagon: (5 − 2) × 180° = 540°
Exterior angles
Walk around any polygon and you turn a full circle. So the exterior angles always add up to 360°.
For a regular polygon, each exterior angle = 360° ÷ n
For a regular polygon, each exterior angle = 360° ÷ n
Interior and exterior together
At each corner, interior + exterior = 180° (they sit on a straight line).
Regular hexagon: exterior = 360° ÷ 6 = 60°, interior = 180° − 60° = 120°
Regular hexagon: exterior = 360° ÷ 6 = 60°, interior = 180° − 60° = 120°
Find the interior angle of a regular octagon.
- Exterior angle = 360° ÷ 8 = 45°
- Interior angle = 180° − 45° = 135°
- Check: sum = (8 − 2) × 180° = 1080°, and 8 × 135° = 1080°. It matches.
Answer: 135°
Interior angles add to (n − 2) × 180°. Exterior angles always add to 360°. Interior + exterior = 180°.
The interactive lesson includes the diagrams for this topic.
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