Regular Polygon Calculator
Give the regular polygon calculator the number of sides and one measurement, and it returns every other length, the area and the angles.
Area
- Side length
- 10
- Apothem (inradius) The distance from the center to the middle of a side, and the radius of the largest circle inside.
- 12.0711
- Circumradius The distance from the center to a corner, and the radius of the circle through every corner.
- 13.0656
- Perimeter
- 80
- Interior angle
- 135°
- Exterior angle
- 45°
- Sum of the interior angles
- 1,080°
- Diagonals
- 20
- Working
- Area = ½ × perimeter × apothem = ½ × 80 × 12.0711 = 482.843
How the Regular Polygon Calculator works
A regular polygon has equal sides and equal angles, so the number of sides n and one length fix every other measurement. Lines from the center to each corner cut it into n identical isosceles triangles, and half of one of them is a right triangle with an angle of 180° ÷ n at the center. That triangle gives the three lengths: for side s, the apothem a = s ÷ (2 tan(180° ÷ n)) and the circumradius R = s ÷ (2 sin(180° ÷ n)).
The area of a regular polygon formula is area = ½ × perimeter × apothem, which is the n triangles added together, each with base s and height a. In terms of the side alone it is n × s² ÷ (4 tan(180° ÷ n)).
Worked example, a regular octagon with sides of 10: tan 22.5° = 0.414214, so the apothem is 10 ÷ 0.828427 = 12.0711. The perimeter is 80, and the area is ½ × 80 × 12.0711 = 482.843. The circumradius is 10 ÷ (2 × sin 22.5°) = 13.0656.
The formula for interior angles of a polygon is (n - 2) × 180° for their sum, because the shape splits into n - 2 triangles from any one corner. Each interior angle of a regular polygon is that sum divided by n: (8 - 2) × 180° = 1,080°, so 135° at each corner of the octagon. The exterior angles always total 360°, so each is 360° ÷ n, 45° here.
Regular Polygon Calculator FAQ
How do I find the side length from the area?
Rearrange the area formula: s = √(4 × area × tan(180° ÷ n) ÷ n). For a regular pentagon of area 100, tan 36° = 0.726543, so s = √(4 × 100 × 0.726543 ÷ 5) = 7.62387.
How many diagonals does a polygon have?
n(n - 3) ÷ 2. Each corner connects to every other corner except itself and its two neighbors, and each diagonal is counted from both ends. An octagon has 8 × 5 ÷ 2 = 20.
What happens as the number of sides grows?
The polygon approaches its circumscribed circle. A regular 100-gon with circumradius 1 has an area of 3.13953, within 0.07% of π.