What is a polygon?
A polygon is a closed shape with straight sides. "Poly" means many, "gon" means angle.
| Sides | Name | Sides | Name |
|---|---|---|---|
| 3 | triangle | 7 | heptagon |
| 4 | quadrilateral | 8 | octagon |
| 5 | pentagon | 9 | nonagon |
| 6 | hexagon | 10 | decagon |
| 12 | dodecagon | n | n-gon |
Regular and irregular
A regular polygon has all sides equal and all angles equal: a square, an equilateral triangle, a stop sign (regular octagon). Irregular polygons don’t.
Convex and concave
- Convex: every interior angle is less than 180°. All diagonals stay inside.
- Concave: at least one interior angle is more than 180°, so the shape "caves in" (like an arrowhead).
Sum of interior angles
Pick one vertex and draw diagonals to every other vertex. An n-sided polygon splits into (n − 2) triangles. So:
Sum of interior angles = (n − 2) × 180°
| Polygon | n | Triangles | Angle sum |
|---|---|---|---|
| Triangle | 3 | 1 | 180° |
| Quadrilateral | 4 | 2 | 360° |
| Pentagon | 5 | 3 | 540° |
| Hexagon | 6 | 4 | 720° |
| Octagon | 8 | 6 | 1,080° |
Each interior angle of a regular polygon
Divide the sum by n.
- Regular hexagon: 720° ÷ 6 = 120°
- Regular octagon: 1,080° ÷ 8 = 135°
Exterior angles
An exterior angle is formed by extending one side. For any convex polygon, the exterior angles add to 360° (you turn one full turn walking around it).
For a regular polygon, each exterior angle = 360° ÷ n.
- Regular pentagon: 360 ÷ 5 = 72°, so each interior angle = 180 − 72 = 108°.
Working backwards
Example: Each interior angle of a regular polygon is 140°. How many sides?
- Exterior angle = 180 − 140 = 40°
- n = 360 ÷ 40 = 9 (a nonagon)
Tessellations
Regular polygons that tile a flat surface with no gaps: triangles (60°), squares (90°) and hexagons (120°). Their angles divide 360° evenly. Bees use hexagons in honeycombs!
Worked examples
Example 1: Sum of interior angles of a decagon?
(10 − 2) × 180 = 1,440°.
Example 2: Each exterior angle of a regular 12-gon?
360 ÷ 12 = 30°.
Example 3: Can a regular polygon have exterior angles of 50°?
360 ÷ 50 = 7.2 — not a whole number. No.
Common mistakes
- Using n × 180 instead of (n − 2) × 180.
- Thinking exterior angles add to 180°. They add to 360°.
- Dividing the angle sum by n for irregular polygons. That only works when all angles are equal.
Try it yourself
- Sum of interior angles of a heptagon?
- Each interior angle of a regular decagon?
- A regular polygon has exterior angles of 24°. How many sides?
- Five angles of a hexagon are 110°, 130°, 120°, 125° and 115°. Find the sixth.
Answers: 1) 900° 2) 144° 3) 15 4) 120°
Quick recap
- Interior angle sum = (n − 2) × 180°.
- Exterior angles of a convex polygon add to 360°.
- A regular polygon has equal sides and angles: each exterior angle = 360° ÷ n.