Polygon Calculator

A polygon is a 2D closed figure made up of straight line segments. That's the polygon definition. But what does it look like? Many shapes you learned about are polygons - triangles, squares, parallelograms, rhombus, kites, pentagons, hexagons, octagons... A lot of them. But popular shapes that are not polygons also exist - take a circle and an ellipsis as an example.

Polygons are classified based on their sides and angles, as well as their convexity, symmetry, and other properties.

In this polygon calculator, we solve the properties of regular polygons - the special polygon types which are:

• Equiangular — all angles are equal in measure
• Equilateral — all sides have the same length

Every regular polygon with n sides is formed by n isosceles triangles.

The answer to the question depends on which polygon you have on your mind. "Usually, you can use the polygon name as a hint to deduce how many sides it has - the prefixes come from Greek numbers.

If your polygon has 11 or more sides, it's easier to write 11-gon, 14-gon, 20-gon ... "100-gon", etc. But if you're really curious...

If you want to calculate the regular polygon parameters directly from equations, all you need to know is the polygon shape and its side length:

1. Area

• area = n × a² × cot(π/n)/ 4

Where n- number of sides, a - side length

Other equations, which use parameters such as the circumradius or perimeter, can also be used to determine the area. You can find them in a dedicated calculator of polygon area.

2. Perimeter

• perimeter = n × a

Read more about polygon perimeter in the perimeter of a polygon calculator.

3. Angles :

• α = (n - 2) × π / n, where α is an interior angle;
• β = 2 × π / n, where β is an exterior angle.

If you're particularly interested in angles, you may want to take a look at our polygon angle calculator.

4. Incircle radius (apothem)

• ri = a / (2 × tan(π/n))

5. Circumcircle radius

• rc = a / (2 × sin(π/n))

All these equations are implemented in our polygon calculator.

If you're still wondering how to use our tool, have a look at the following example:

1. Choose the polygon shape and type its number of sides. To calculate the properties of, e.g., a nonagon, type 9 into the number of sides box.
2. Enter one parameter. One given value is enough. Assume that we know the perimeter of our shape; let's say it's 18 in.
3. Great! Our polygon calculator finds all the remaining values! We determined that:
   • side = 2 in
   • area = 24.727 in²
   • α = 140°
   • β = 40°
   • rc = 2.924 in
   • ri = 2.7475 in