What this calculator does
A regular dodecagon is a twelve-sided polygon with every side and every interior angle equal, the shape behind some coins, clock faces and architectural details. This calculator takes any one of three common measurements, the side length, the apothem (the distance from the centre to the midpoint of a side) or the circumradius (the distance from the centre to a corner), and works out the area, perimeter and the other two measurements from it.
The area formula for a regular dodecagon, 3 × (2 + √3) × s², follows the same pattern used for every regular polygon: split the shape into twelve identical triangles meeting at the centre, and multiply the area of one triangle by twelve. The constant 2 + √3 is simply cot(15°) tidied into a closed form, since 360° ÷ 12 gives a 15° angle at the centre of each triangle.
The formula
Whichever measurement is entered is first converted to the side length, since every other figure is derived from that. The apothem and circumradius both relate to the side length through the half-angle at the centre, π/12 radians (15°). Once the side length is known, area follows from 3 × (2 + √3) × s² and perimeter from 12 × s.
| Term | Meaning |
|---|---|
| Side length (s) | The length of one of the dodecagon's twelve equal edges. |
| Apothem | The perpendicular distance from the centre to the midpoint of a side, also called the inradius. |
| Circumradius | The distance from the centre to any one of the twelve corners. |
The inputs explained
| Field | What to enter |
|---|---|
| Value | Enter the measurement you know: the side length, apothem or circumradius, whichever is easiest to measure. |
| This value is the | Tell the calculator which of the three measurements the value above actually is. |
When to use it
Costing dodecagonal paving or tiling
A twelve-sided paving stone or tile's area, worked out from a single measured side, gives the material needed per piece before multiplying up by however many are being laid.
Designing from a fixed diameter
If a dodecagon needs to fit within a circle of a known size, such as a coin blank, that circle's radius is the circumradius, and this calculator converts it straight to the side length and area.
Checking a geometry problem
Interior angle, area and perimeter of a regular dodecagon are common exam questions; this calculator confirms the arithmetic behind the 3 × (2 + √3) × s² formula.
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 does dodecagon area change with side length?
A regular dodecagon's area and perimeter across a spread of side lengths.
| Side length | Area | Perimeter |
|---|---|---|
| 5 | 279.904 | 60.000 |
| 8 | 716.554 | 96.000 |
| 10 | 1,119.62 | 120.000 |
| 12 | 1,612.25 | 144.000 |
| 15 | 2,519.13 | 180.000 |
| 20 | 4,478.46 | 240.000 |
Questions
What is the interior angle of a regular dodecagon?
Each interior angle is 150°, following the general regular-polygon formula (n − 2) × 180° ÷ n, applied with n = 12.
What real objects are dodecagon-shaped?
The UK's current £1 coin is a well-known regular dodecagon, along with some clock and gauge faces and architectural window and floor patterns, chosen partly because a twelve-sided shape is hard to counterfeit by flattening a circle and easy to divide into clean angular segments.
Does this work for an irregular twelve-sided shape?
No. This calculator assumes every side and angle is equal, which is what makes a single side length (or apothem, or circumradius) enough to describe the whole shape. An irregular dodecagon needs its individual sides and angles measured separately.
How is the apothem different from the circumradius?
The apothem runs from the centre to the middle of an edge, while the circumradius runs from the centre to a corner; the circumradius is always the longer of the two, since a corner sits further from the centre than the midpoint of the side next to it.
For the six-, seven- and eight-sided versions of the same calculation, see the hexagon calculator, heptagon calculator and octagon calculator. For a regular polygon with any number of sides, use the regular polygon calculator.