Pentagon Calculator
What's the pentagon definition? How many sides does a pentagon have? Find the answers with our pentagon calculator!
Working
| Sides | 5 (pentagon) |
| Interior angle sum | 540° |
| Area from apothem | ½ × perimeter × apothem = 61.937186 |
| Circumcircle area | 81.838001 |
| Polygon fills | 75.683% of its circumcircle |
The formula
A = n·s² ÷ (4·tan(π/n)), interior angle = (n−2)·180° ÷ n
Two ways to see the area
Split a regular n-gon into n triangles from the centre. Each has base s and height equal to the apothem, so the total area is ½ × perimeter × apothem — the same shape of formula as a circle’s ½ × circumference × radius. As n grows, the polygon becomes a circle and the two formulas converge.
Area scales with the square
Double every length in a flat shape and the area quadruples; triple them and it grows ninefold. Perimeter, being a length, scales in step with the sides. That mismatch is why bigger shapes enclose proportionally more area per unit of edge — and why a circle, having no corners at all, encloses the most area for a given perimeter of any shape.
Units
Whatever unit you enter, the area comes out in that unit squared and the perimeter in the unit itself. The calculator is unit-agnostic — enter centimetres and read cm and cm², or feet and read ft and ft². Just do not mix units between fields.