What this calculator does
A regular polygon has every side and every angle equal, which means it is fully described by just two numbers: how many sides it has, and how long each one is. Everything else, area, interior angle, the circles that touch its corners and its edges, follows from those two alone.
The shape can be split into n identical isosceles triangles meeting at the centre, and that construction is where every formula here comes from. The angle at the centre of each triangle is 360° divided by n, and the triangle’s own geometry supplies the rest.
The formula
The circumradius is the distance from the centre to a corner: the triangle’s two equal sides. The inradius, or apothem, is the distance from the centre to the middle of a side: the triangle’s height. Area is n times the area of one of those triangles, which simplifies to the formula shown. As n grows large, a regular polygon approaches a circle, and the inradius and circumradius converge toward the same value, the radius of that circle.
| Term | Meaning |
|---|---|
| n | The number of sides. |
| s | The length of each side. |
| Circumradius | Centre to a corner: the radius of the circle passing through every vertex. |
| Inradius (apothem) | Centre to the middle of a side: the radius of the largest circle that fits inside. |
| Interior angle | The angle at each corner, equal to (n − 2) × 180° ÷ n. |
The inputs explained
| Field | What to enter |
|---|---|
| Number of sides | The number of sides. 3 is a triangle, 4 a square, 6 a hexagon and so on. |
| Side length | The length of one side: every side is the same in a regular polygon. |
When to use it
Laying out a hexagonal or octagonal feature
Paving, gazebos, planter boxes and stop signs are commonly regular polygons. The circumradius tells you the diameter of circle needed to mark out the corners on site.
Cutting stock material into a polygon
The apothem, the inradius, is the distance from the centre to the flat of each side, which is the figure needed when a polygon must fit inside a circular blank of known size.
Estimating the area of a many-sided shape
A shape with a large number of short, equal sides is well approximated as a regular polygon, and the area formula here converges toward πr² as n increases, matching a circle of that radius.
Working out bolt-hole or spoke spacing
Equally spaced points around a circle, bolt patterns, wheel spokes, clock numerals, form a regular polygon, and the interior and exterior angles describe the spacing between them.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
From a triangle to a near-circle, side fixed at 10
The same side length carried across shapes with more and more sides.
| Sides | Area | Interior angle | Circumradius |
|---|---|---|---|
| 3 | 43.301 | 60.00° | 5.774 |
| 4 | 100.000 | 90.00° | 7.071 |
| 5 | 172.048 | 108.00° | 8.507 |
| 6 | 259.808 | 120.00° | 10.000 |
| 8 | 482.843 | 135.00° | 13.066 |
| 12 | 1,119.62 | 150.00° | 19.319 |
A hexagon growing in size
A fixed 6-sided shape scaled up through a range of side lengths.
| Side length | Area | Perimeter | Circumradius |
|---|---|---|---|
| 1 | 2.598 | 6.000 | 1.000 |
| 5 | 64.952 | 30.000 | 5.000 |
| 10 | 259.808 | 60.000 | 10.000 |
| 20 | 1,039.23 | 120.000 | 20.000 |
| 50 | 6,495.19 | 300.000 | 50.000 |
Questions
What is the difference between the inradius and circumradius?
The circumradius reaches the corners; the inradius (or apothem) reaches the middle of a side. The circumradius is always the larger of the two for any polygon with three or more sides.
How do I find the interior angle of a regular polygon?
Subtract 2 from the number of sides, multiply by 180°, then divide by the number of sides. A hexagon, for example, gives (6 − 2) × 180 ÷ 6 = 120°.
Why does a hexagon’s circumradius equal its side length?
Because a regular hexagon is made of six equilateral triangles meeting at the centre, each with all three sides equal: including the two that are circumradii. No other regular polygon has this property.
What happens to a regular polygon as the number of sides increases?
It approaches a circle. The interior angle climbs toward 180°, and the area formula converges on πr², with the inradius and circumradius both tending to that same radius.
Does this work for irregular polygons?
No: every formula here assumes equal sides and equal angles. An irregular polygon needs to be split into triangles and measured piece by piece, for instance with the triangle calculator.
For a polygon with infinitely many sides, see the circle calculator. For an irregular shape, break it into triangles using Heron’s formula.