What this calculator does
A decagon is a 10-sided polygon, and a regular decagon has all 10 sides and all 10 interior angles equal. It shows up in coin and medal shapes, architectural tracery and tiling patterns, and, like any regular polygon, its area, perimeter and radii can all be worked out from a single known measurement.
This calculator accepts the side length directly, or the apothem (the distance from the centre to the middle of a side) or the circumradius (the distance from the centre to a corner), and converts whichever one you enter into all three, along with the interior angle and total area.
The formula
The area of a regular decagon with side length s is (10/4) × s² × cot(π/10). If you enter the apothem or circumradius instead of the side length, the calculator first recovers the side length from that measurement using the same trigonometric relationship, then applies the area formula as normal.
| Term | Meaning |
|---|---|
| Side length (s) | The length of one of the 10 equal edges of the decagon. |
| Apothem | The distance from the centre of the decagon to the midpoint of one side, also the radius of the largest circle that fits inside it. |
| Circumradius | The distance from the centre of the decagon to one of its 10 corners, also the radius of the circle passing through every corner. |
| Interior angle | The angle at each corner of a regular decagon, fixed at 144° regardless of size. |
The inputs explained
| Field | What to enter |
|---|---|
| Value | The measurement you know: the side length, the apothem, or the circumradius. |
| This value is the | Which of those three the value above represents. |
When to use it
Working out material for a decagonal build
A decagonal deck, planter or gazebo base needs the perimeter to order edging and the area to order decking or paving, both of which follow directly from a single side-length measurement.
Checking a design against a known circumradius
Decagon shapes are sometimes specified by the circle they are inscribed in, such as a coin blank cut to a fixed diameter; entering that circumradius converts it straight into the resulting side length and area.
Comparing a decagon to other regular polygons
Interior angle and area-to-perimeter ratio both change with the number of sides; working a decagon through the same calculation as a hexagon or octagon shows how those figures shift as a shape gains sides.
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.
How decagon area and perimeter scale with side length
Area and perimeter for a regular decagon at a range of side lengths.
| Side length | Area | Perimeter |
|---|---|---|
| 5 | 192.355 | 50.000 |
| 10 | 769.421 | 100.000 |
| 15 | 1,731.20 | 150.000 |
| 20 | 3,077.68 | 200.000 |
| 30 | 6,924.79 | 300.000 |
| 50 | 19,235.52 | 500.000 |
Questions
How many sides does a decagon have?
10. The name comes from the Greek deka, meaning ten, the same root as decade.
What is the interior angle of a regular decagon?
144°, found from (n − 2) × 180° ÷ n with n = 10, which gives (10 − 2) × 180 ÷ 10 = 144°.
What is the formula for the area of a regular decagon?
Area = (10/4) × s² × cot(π/10), where s is the side length. Entering an apothem or circumradius instead first converts it back to a side length using the same geometry.
How is a decagon different from a dodecagon?
A decagon has 10 sides and a 144° interior angle; a dodecagon has 12 sides and a 150° interior angle. Both are regular polygons and follow the same style of area, perimeter and angle formulas, just with a different number of sides.
For other regular polygons, see the octagon calculator and the dodecagon calculator, or the general regular polygon calculator for any number of sides.