Polygon
Regular polygon area and perimeter
About This Calculator
A regular polygon has all sides equal and all interior angles equal. From triangles to hexagons to dodecagons, regular polygons appear in architecture, tiling, and engineering. As the number of sides increases, a regular polygon approaches a circle.
Formula
Perimeter = n * side
Area = (n * side^2) / (4 * tan(pi/n))
Interior angle = (n-2) * 180 / n degrees
Example Calculation
Regular hexagon with side length 6
- Perimeter = 6 * 6 = 36
- Area = (6 * 36) / (4 * tan(pi/6)) = 216 / (4 * 0.5774) = 216 / 2.309 = 93.53
- Interior angle = (6-2) * 180 / 6 = 120°
Perimeter=36; Area≈93.53; Interior angle=120°
Regular Polygon Properties
| Shape | Sides | Interior Angle | Area (side=1) |
|---|---|---|---|
| Triangle | 3 | 60° | 0.433 |
| Square | 4 | 90° | 1.000 |
| Pentagon | 5 | 108° | 1.720 |
| Hexagon | 6 | 120° | 2.598 |
| Octagon | 8 | 135° | 4.828 |
| Decagon | 10 | 144° | 7.694 |
| Dodecagon | 12 | 150° | 11.196 |
Frequently Asked Questions
Why do bees build hexagonal honeycombs?
Hexagons are the most efficient shape for tiling a flat surface — they minimize the total perimeter (wax used) for a given area, and they fit together without gaps. This makes hexagonal cells optimal for storing honey.
What is the sum of interior angles of any polygon?
For a polygon with n sides, the sum of interior angles = (n-2) * 180 degrees. For a triangle (n=3): 180°; square (n=4): 360°; hexagon (n=6): 720°.
What is an apothem?
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of one of its sides. Area = (1/2) * perimeter * apothem.