Perimeter of a Polygon Calculator
Enter the number of sides and any one dimension of a regular polygon, and this calculator instantly finds the perimeter, area, inradius, circumradius, and both interior and exterior angles. Switch between metres, centimetres, feet, and inches. The step-by-step panel shows the exact formulas with your numbers substituted in.
What is the perimeter of a regular polygon?
The perimeter of a polygon is the total length of its boundary. For a regular polygon, where every side is the same length, the formula is simply P = n x a, where n is the number of sides and a is the side length. For example, a regular hexagon with each side 10 cm long has a perimeter of 6 x 10 = 60 cm. For irregular polygons, you add up all the individual side lengths since they differ from one another.
Inradius, circumradius, and how they relate to the side
A regular polygon can be described by three circles. The incircle (radius r, the inradius or apothem) fits perfectly inside the polygon touching each side at its midpoint. The circumcircle (radius R, the circumradius) passes through every vertex. Given the side length a and the number of sides n, the relationships are: r = a / (2 x tan(pi / n)) and R = a / (2 x sin(pi / n)). This calculator lets you enter any one of these values and computes the others, so you can solve the polygon from whichever dimension you already know.
Interior and exterior angles of a regular polygon
Every interior angle of a regular n-gon measures (n - 2) x 180 / n degrees. The exterior angle, formed by extending one side, is 360 / n degrees. The two always add to 180 degrees. For a triangle (n = 3) the interior angle is 60 degrees; for a square (n = 4) it is 90 degrees; for a hexagon (n = 6) it is 120 degrees. As n grows very large, the polygon approaches a circle and the interior angle approaches 180 degrees.
Area of a regular polygon
The area of a regular polygon is A = (n x a^2) / (4 x tan(pi / n)). This formula is equivalent to dividing the polygon into n identical isosceles triangles from the centre, computing each triangle's area as (1/2) x base x height where the base is the side length a and the height is the inradius r, then multiplying by n. Because of this, it can also be written as A = (1/2) x n x a x r. This calculator derives the area regardless of which input you start from, and shows every step of the working.
Regular polygon properties reference
| Sides (n) | Name | Interior angle | Exterior angle | Area formula |
|---|---|---|---|---|
| 3 | Triangle | 60 deg | 120 deg | (a^2 * sqrt(3)) / 4 |
| 4 | Square | 90 deg | 90 deg | a^2 |
| 5 | Pentagon | 108 deg | 72 deg | (a^2 / 4) * sqrt(25 + 10*sqrt(5)) |
| 6 | Hexagon | 120 deg | 60 deg | (3*sqrt(3) / 2) * a^2 |
| 7 | Heptagon | 128.57 deg | 51.43 deg | (7 * a^2) / (4 * tan(pi/7)) |
| 8 | Octagon | 135 deg | 45 deg | 2 * (1 + sqrt(2)) * a^2 |
| 9 | Nonagon | 140 deg | 40 deg | (9 * a^2) / (4 * tan(pi/9)) |
| 10 | Decagon | 144 deg | 36 deg | (5 * a^2 / 2) * sqrt(5 + 2*sqrt(5)) |
| 12 | Dodecagon | 150 deg | 30 deg | 3 * (2 + sqrt(3)) * a^2 |
Interior angle, exterior angle, and formulas for common regular polygons.
Frequently asked questions
How do I find the perimeter of a regular polygon?
Multiply the number of sides by the side length: P = n x a. For a regular pentagon with side length 8 cm: P = 5 x 8 = 40 cm. If you only know the inradius, circumradius, or area instead, this calculator derives the side length first and then the perimeter.
What is the difference between inradius and circumradius?
The inradius (also called the apothem) is the distance from the centre of the polygon to the midpoint of any side. The circumradius is the distance from the centre to any vertex. For a given regular polygon, the circumradius is always larger than the inradius. They are related to the side length by r = a / (2 x tan(pi / n)) and R = a / (2 x sin(pi / n)).
Can I use this calculator for irregular polygons?
This calculator is designed for regular polygons, where all sides and all angles are equal. For an irregular polygon, the perimeter is simply the sum of all individual side lengths. You would need to measure each side separately and add them together.
What is the interior angle of a regular hexagon?
A regular hexagon has 6 sides. Its interior angle is (6 - 2) x 180 / 6 = 120 degrees. The exterior angle is 360 / 6 = 60 degrees. The sum of all interior angles of a hexagon is 720 degrees.
Does the number of sides have to be a whole number?
Yes. A polygon must have a whole number of sides (at least 3). You cannot have a polygon with 5.5 sides. This calculator rounds any decimal entry to the nearest integer to guard against rounding errors.
How do I find the perimeter if I only know the area?
Rearrange the area formula A = (n x a^2) / (4 x tan(pi / n)) to solve for a: a = sqrt(4 x A x tan(pi / n) / n). Then the perimeter is P = n x a. This calculator does that automatically when you select "Area" as the known dimension.