Polygons – Shapes with Any Number of Sides
A polygon is any closed, flat shape made up of straight sides. Triangles and quadrilaterals are polygons, but polygons can have any number of sides from three upward. Understanding their properties opens the door to advanced geometry.
What Is a Polygon?
A polygon is a closed two-dimensional shape with three or more straight sides. It is regular if all its sides are equal AND all its angles are equal. Otherwise it is irregular.
Common Polygons
| Sides | Name | Interior Angle (regular) |
|---|---|---|
| 3 | Triangle | 60° |
| 4 | Quadrilateral | 90° |
| 5 | Pentagon | 108° |
| 6 | Hexagon | 120° |
| 7 | Heptagon | 128.6° |
| 8 | Octagon | 135° |
| 10 | Decagon | 144° |
| 12 | Dodecagon | 150° |
| n | n-gon | (n − 2) × 180 / n |
Interior Angle Sum
Sum of interior angles of an n-sided polygon = (n − 2) × 180°.
(6 − 2) × 180 = 4 × 180 = 720°.
Each interior angle = (n − 2) × 180 / n = 140. Solve: 180n − 360 = 140n. 40n = 360. n = 9 sides (a nonagon).
Exterior Angles
The exterior angles of any polygon (one at each vertex) always sum to 360°, regardless of the number of sides. For a regular polygon, each exterior angle = 360 / n.
360 / 8 = 45°. Check: interior angle = 180 − 45 = 135° = (8−2)×180/8. Correct.
Number of Diagonals
An n-sided polygon has n(n − 3) / 2 diagonals.
Key Takeaways
- A polygon is closed, flat, and made of straight sides.
- Interior angle sum = (n − 2) × 180°.
- Exterior angles always sum to 360° for any polygon.
- A regular polygon has all sides equal and all angles equal.
Practice Questions
- Find the sum of interior angles of a decagon (10 sides).
- Each interior angle of a regular polygon is 150°. How many sides does it have?
- Find the exterior angle of a regular pentagon.
- A polygon has interior angle sum of 1260°. How many sides does it have?
- How many diagonals does a hexagon have?