Regular Polygon Calculator
The area, perimeter, apothem, circumradius and angles of a regular polygon from its side, apothem or circumradius.
Result
Area: 2.5981 ft²
A measurement, not a material estimate. Nothing is added for waste or cuts.
- Perimeter
- 6 ft (72 in, 1.8288 m)
- Side length
- 1 ft (12 in, 0.3048 m)
- Apothem
- 0.866 ft (10.3923 in, 0.264 m)
- Circumradius
- 1 ft (12 in, 0.3048 m)
- Interior angle
- 120 °
- Exterior angle
- 60 °
- Sum of the interior angles
- 720 °
- Number of diagonals
- 9
https://calcgallery.com/geometry/regular-polygon-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Regular Polygon Calculator
- Enter the number of sides.
- Choose what you measured. The side length, the apothem from the centre to the middle of a side, or the circumradius from the centre to a corner.
- Enter that length and read the area, then every other measurement below it.
What makes a polygon regular
A regular polygon has all its sides the same length and all its corners the same angle, like an equilateral triangle, a square, a stop sign or a honeycomb cell. Because it is so symmetric, a single length and the number of sides fix everything else.
The formulas
For n sides of length s:
| Measurement | Formula |
|---|---|
| Perimeter | n × s |
| Apothem | s ÷ (2 tan(π ÷ n)) |
| Circumradius | s ÷ (2 sin(π ÷ n)) |
| Area | ½ × perimeter × apothem = ¼ n s² cot(π ÷ n) |
| Interior angle | (n − 2) × 180° ÷ n |
| Exterior angle | 360° ÷ n |
| Sum of the interior angles | (n − 2) × 180° |
| Diagonals | n(n − 3) ÷ 2 |
The area is easiest to see from the apothem. Lines from the centre to each corner cut the polygon into n identical triangles. Each has a side as its base and the apothem as its height, so together they give ½ × perimeter × apothem.
Apothem and circumradius
Both are measured from the centre. The apothem goes to the middle of a side and the circumradius to a corner, so the circumradius is always the longer of the two. When setting out an octagonal gazebo or a hexagonal bed from a centre stake, the circumradius tells you where the corners go. When fitting a shape between two parallel lines, twice the apothem is the width across the flats. The page accepts either, or the side, and works out the rest.
The angles
The exterior angles, the turn you make at each corner walking round the outside, always add up to one full turn of 360 degrees, so each is 360° ÷ n. The interior angle is what remains of a straight line, 180° minus the exterior. For a hexagon that is 120 degrees, and for an octagon 135 degrees, which is why octagonal frames are cut with a 22.5 degree mitre on each end.
Material estimates for decking or paving laid in a polygon need an allowance for cuts that this page does not add.
With your numbers
- Side = 1 ft, apothem = s ÷ (2 tan(π ÷ 8)) = 1.2071 ft
- Perimeter = 8 × 1 ft = 8 ft
- Area = ½ × perimeter × apothem = 4.828 ft²
Quick reference
A measurement, not a material estimate. Nothing is added for waste or cuts.
| Example | Area |
|---|---|
| Triangle, 2 m sides | 1.7321 m² |
| Square, 2 m sides | 4 m² |
| Pentagon, 5 m sides | 43.0119 m² |
| Hexagon, 2 m sides | 10.3923 m² |
| Octagon, 1 ft sides | 4.8284 ft² |
| Hexagon, 6 ft across the corners | 23.3827 ft² |
| Octagon, 10 ft across the flats | 82.8427 ft² |
| Dodecagon, 1 m sides | 11.1962 m² |
Common questions about the Regular Polygon Calculator
- What is an apothem?
- The distance from the centre of a regular polygon straight to the middle of one side, meeting the side at a right angle. It is the radius of the largest circle that fits inside. For a hexagon or octagon, twice the apothem is the distance across the flats, which is what a spanner or a nut size measures.
- How do I lay out a hexagon?
- A regular hexagon’s circumradius equals its side. Draw a circle with a radius equal to the side you want. Then, keeping the compass or string at the same length, step round the circle marking six points, and join them. For a hexagonal patio or deck, drive a centre stake and use a string line of that length.
- Why does a polygon approach a circle as the sides increase?
- With more and shorter sides, each corner is flatter and the outline hugs the circle through the corners more closely. A 1,000-sided polygon with a circumradius of 1 has an area within 0.00003 of π. Archimedes used the same idea with 96 sides to show that π lies between 3 10/71 and 3 1/7.
- How many diagonals does a polygon have?
- n(n − 3) ÷ 2. From each corner you can draw a diagonal to every other corner except itself and its two neighbours, which is n − 3 diagonals. That counts each diagonal twice, once from each end, hence the halving. A triangle has none, a square 2 and a hexagon 9.