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.
The word “polygon” comes from the Greek poly (many) and gonia (angle). Regular polygons fascinated the ancient Greeks, who wondered which ones could be drawn using only a compass and an unmarked straightedge; it took over 2,000 years to fully answer the question, until a 19-year-old Carl Friedrich Gauss proved in 1796 that a regular 17-sided polygon (a heptadecagon) could be constructed this way — a discovery he was so proud of that he reportedly asked for the shape to be carved on his gravestone. Regular hexagons appear throughout nature for a practical reason: bees build honeycomb cells as hexagons because that shape tiles a plane perfectly with no wasted space, using the least wax to store the most honey. A modern football (soccer ball) is a more complex example — its classic pattern is a truncated icosahedron, stitched together from 12 regular pentagons and 20 regular hexagons.
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: Angle Sums & Sides
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