Regular polygons
A regular polygon has all sides equal and all angles equal: the equilateral triangle, the square, the regular pentagon, the hexagon of a honeycomb or a nut, the octagon of a stop sign. Because of that symmetry, knowing the number of sides and one side length is enough to work out everything else.
The formulas
For n sides of length s (angles in degrees):
- Perimeter = n × s
- Apothem (distance from the center to the middle of a side) = s ÷ (2 × tan(180° ÷ n))
- Area = perimeter × apothem ÷ 2 = n × s × apothem ÷ 2
- Circumradius (center to a corner) = s ÷ (2 × sin(180° ÷ n))
- Interior angle = (n − 2) × 180° ÷ n, and the exterior angle is 360° ÷ n
- Number of diagonals = n × (n − 3) ÷ 2
Worked example: a hexagon with 10 cm sides
- Perimeter: 6 × 10 = 60 cm.
- Apothem: 10 ÷ (2 × tan 30°) = 8.66 cm.
- Area: 60 × 8.66 ÷ 2 = 259.81 cm².
- Circumradius: 10 ÷ (2 × sin 30°) = 10 cm. In a regular hexagon the radius equals the side, which is why six equilateral triangles tile it.
- Interior angle: 4 × 180 ÷ 6 = 120°; there are 9 diagonals.
- Across corners the hexagon measures 20 cm; across flats, 2 × 8.66 = 17.32 cm, the size a wrench must fit.
Same 10 cm side, other polygons
| Polygon | Sides | Area | Interior angle | Circumradius |
|---|---|---|---|---|
| Triangle | 3 | 43.30 | 60° | 5.774 |
| Square | 4 | 100.00 | 90° | 7.071 |
| Pentagon | 5 | 172.05 | 108° | 8.507 |
| Hexagon | 6 | 259.81 | 120° | 10.000 |
| Octagon | 8 | 482.84 | 135° | 13.066 |
| Decagon | 10 | 769.42 | 144° | 16.180 |
| Dodecagon | 12 | 1,119.62 | 150° | 19.319 |
As the number of sides grows, the polygon approaches a circle, and its area approaches π × r².
Practical uses
- Woodworking: an octagonal table or frame needs miter cuts at 22.5°, half the exterior angle of 45°.
- Gazebos and planters: the circumradius tells you the footprint, the apothem the distance from the center to each wall.
- Tiling: only triangles, squares and hexagons tile a floor on their own, because their interior angles divide 360° exactly.
- Pentagon and the golden ratio: in a regular pentagon, a diagonal divided by a side equals φ ≈ 1.618.