CalcKit
Polygon Calculator
Table of Contents
Polygon definition
A polygon is a flat, closed shape made of straight sides. The sides meet in pairs at the corners, called vertices, and a polygon always has as many vertices as it has sides.
This tool works with the regular polygon, whose sides all have the same length and whose angles are all equal. The equilateral triangle, the square and the regular hexagon are all regular polygons.
A regular polygon with n sides has the following properties:
- All n sides are the same length
- Each interior angle is
(n - 2) × 180° / n - The interior angles add up to
(n - 2) × 180° - All its vertices lie on one circle (the circumcircle), and one circle (the incircle) touches all its sides
- It can be divided into n identical isosceles triangles meeting at its centre
The more sides a regular polygon has, the closer it looks to a circle.
Parts of a regular polygon
The six parts of a regular polygon mentioned in this tool are the number of sides (n), side length (a), area (A), perimeter (P), apothem (r) and radius (R).
Number of sides
The number of sides (n) is how many sides the polygon has. It is at least 3, for a triangle.
It can be found with the following formulas:
-
If you know the perimeter and the side length:
n = P / a -
If you know the area, the apothem and the side length:
n = 2 * A / (r * a)
Side length (a)
The side length is the length of one side of the polygon.
It can be found with the following formulas:
-
If you know the perimeter and the number of sides:
a = P / n -
If you know the apothem and the number of sides:
a = 2 * r * tan(180° / n) -
If you know the radius and the number of sides:
a = 2 * R * sin(180° / n)
Area (A)
The area of a regular polygon is the space enclosed by its sides.
It can be found with the following formulas:
-
If you know the perimeter and the apothem:
A = P * r / 2 -
If you know the number of sides and the radius:
A = n * R² * sin(360° / n) / 2 -
If you know the number of sides and the side length:
A = n * a² / (4 * tan(180° / n))
Perimeter (P)
The perimeter of a regular polygon is the total length of its sides:
P = n * a
Apothem (r)
The apothem is the distance from the centre of the polygon to the middle of one of its sides. It is also the radius of the inscribed circle (the inradius), the largest circle that fits inside the polygon.
It can be found with the following formulas:
-
If you know the radius and the side length:
r = √(R² - a² / 4) -
If you know the side length and the number of sides:
r = a / (2 * tan(180° / n))
Radius (R)
The radius (or circumradius) is the distance from the centre of the polygon to one of its vertices. It is also the radius of the circle that passes through all the vertices.
It can be found with the following formulas:
-
If you know the apothem and the side length:
R = √(r² + a² / 4) -
If you know the side length and the number of sides:
R = a / (2 * sin(180° / n))
The radius, the apothem and half a side form a right triangle, which is where most of these formulas come from.
Names of polygons
Polygons are named after their number of sides:
- 3 sides: triangle
- 4 sides: quadrilateral (a regular one is a square)
- 5 sides: pentagon
- 6 sides: hexagon
- 7 sides: heptagon
- 8 sides: octagon
- 9 sides: nonagon
- 10 sides: decagon
- 12 sides: dodecagon
A polygon is convex when all its interior angles are less than 180°, and concave when at least one is larger. Every regular polygon is convex.
Frequently asked questions
How do you find the area of a regular polygon?
Multiply the perimeter by the apothem and divide by two: A = P * r / 2. If you only know the side length, use A = n * a² / (4 * tan(180° / n)). A regular octagon with a = 2 has an area of 8 × 2² / (4 × tan(22.5°)) ≈ 19.31 square units.
What is the apothem of a polygon?
The apothem is the distance from the centre of a regular polygon to the middle of a side, at a right angle to that side. From the side length: r = a / (2 * tan(180° / n)).
How do you find the interior angle of a regular polygon?
Use (n - 2) × 180° / n. A regular pentagon has interior angles of 108°, a hexagon 120° and an octagon 135°.
What is the difference between the apothem and the radius?
The apothem goes from the centre to the middle of a side, and the radius from the centre to a vertex. The radius is always the longer of the two: R = √(r² + a² / 4).