Regular Polygon Calculator
This regular polygon calculator solves any shape with equal sides and equal angles, from a triangle to a polygon with a hundred sides. Enter the number of sides and one known value, the side length, circumradius, apothem, or area, and the regular polygon calculator returns the area, side, perimeter, apothem, circumradius, angles, and number of diagonals.
Results
The Regular Polygon Calculator is part of our Geometry collection. Browse more geometry to size, convert, and check your work alongside this tool.
What the regular polygon calculator does
A regular polygon has n equal sides and n equal angles, and every one fits around a center point. Its size can be described by the side, the distance from center to a corner (circumradius), the distance from center to a flat side (apothem), or the area. The regular polygon calculator converts any one of these into all the others using basic trigonometry.
Regular polygon area formula
For n sides of length s:
The number of diagonals is n(n − 3) ÷ 2. When you enter the area, apothem, or circumradius, the calculator first solves for s, then applies the same formulas.
Hexagon calculator example
A regular hexagon with 10 cm sides:
The perimeter is 60 cm, the circumradius equals the side at 10 cm, each interior angle is 120°, and a hexagon has 9 diagonals.
How to use the polygon area calculator
Enter the number of sides: 5 for a pentagon, 6 for a hexagon, 8 for an octagon. Choose which value you know and type it in. Press Calculate to get every measurement. Units carry through, so a side in inches gives an area in square inches.
Reading regular polygon formulas results
As the number of sides grows, a regular polygon approaches a circle: the apothem and circumradius get closer, and the area nears π R². A hexagon is the only common polygon whose circumradius equals its side, which is why hexagons tile so neatly. The interior angle tells you the miter cut for each joint: half of it on each piece.
When to use an octagon calculator or hexagon calculator
- Laying out an octagonal gazebo or deck
- Cutting hexagon tiles or pavers
- Sizing a hex bar or nut across flats
- Finding miter angles for polygon frames
- Geometry homework and exams
Common regular polygon calculator mistakes
- Mixing apothem and circumradius. Flat side versus corner.
- Using width across corners as across flats. They differ.
- Applying formulas to irregular shapes. Sides and angles must be equal.
- Cutting the full interior angle. Each piece gets half the joint angle.
Tips for a hexagon area calculator
- For hexagons, circumradius equals the side.
- Measure across flats for octagons; it is easier.
- Round only the final answer.
Area of a regular polygon from triangles
Every regular polygon splits into n identical isosceles triangles meeting at the center. Each has a base equal to the side and a height equal to the apothem, so the total area is n × ½ × s × a. You can check any result by solving one of those triangles with the triangle area calculator. Half of one triangle is a right triangle with the apothem, half the side, and the circumradius as its sides, which the Pythagorean theorem calculator confirms.
Octagon side length calculator for building projects
Builders usually know the overall width of an octagon, measured across flats. The apothem is half that width, so choose apothem in the calculator and enter half the width. For a gazebo 12 ft across flats, the apothem is 6 ft, each side is 4.97 ft, and each joint is cut at 22.5° on both pieces. For a round feature of the same size, compare with the circle calculator.
Interior angle of a polygon and area for a side of 1
| Polygon (sides) | Interior angle | Area for side = 1 | Apothem for side = 1 |
|---|---|---|---|
| Triangle (3) | 60.00° | 0.4330 | 0.2887 |
| Square (4) | 90.00° | 1.0000 | 0.5000 |
| Pentagon (5) | 108.00° | 1.7205 | 0.6882 |
| Hexagon (6) | 120.00° | 2.5981 | 0.8660 |
| Heptagon (7) | 128.57° | 3.6339 | 1.0383 |
| Octagon (8) | 135.00° | 4.8284 | 1.2071 |
| Nonagon (9) | 140.00° | 6.1818 | 1.3737 |
| Decagon (10) | 144.00° | 7.6942 | 1.5388 |
| Dodecagon (12) | 150.00° | 11.1962 | 1.8660 |
Regular Polygon Calculator FAQ
How do you find the area of a regular polygon?
Area = ½ × perimeter × apothem. A hexagon with 10 cm sides has a 60 cm perimeter and an 8.66 cm apothem, so the area is 259.81 cm².
What is the interior angle of an octagon?
(8 − 2) × 180 ÷ 8 = 135°.
How do I find the side of an octagon from its width?
For a regular octagon, the width across flats is twice the apothem, so s = width × tan(22.5°). A 24 in octagon has sides of about 9.94 in.
What is an apothem?
The distance from the center to the middle of a side, the radius of the inscribed circle.