# Dodecagon Calculator

Created by Anna Szczepanek, PhD
Reviewed by Luis Hoyos
Last updated: Jun 05, 2023

Omni's dodecagon calculator is here to help you answer all the questions related to dodecagons! This tool can work out all the missing values based on just one piece of information, be it the dodecagon diagonal, side, area, perimeter, or incircle/circumcircle radius.

As is our custom in Omni, we also provide a short explanation of the dodecagon formulas implemented in the tool so that you can do the calculations by hand as well. Apart from that, we discuss in detail several most confusing problems related to dodecagons, including:

• How to calculate the sum of the interior angles of a dodecagon?
• How many diagonals does a dodecagon have and how to get this result? (we give two ways!)

To make sure we're on the same page, let us start by recalling what a dodecagon is.

## What's a dodecagon and its main formulas?

A dodecagon is a polygon with twelve sides. The name resembles words like pentagon or octagon, isn't it? This is because it follows the same principle: the prefix is a Greek numeral describing how many sides (or angles) our polygon has. Since dodeka is Greek for twelve, we can now play around with dodecagons. If our dodecagon has all sides equal and all interior angles identical, we call it a regular dodecagon. This is exactly the shape you can see in the picture!

Once we've recalled what a dodecagon is, let's move on to the formulas! From now on, we consider a regular decagon with side length a. Since it has twelve sides, we bet you can guess the dodecagon perimeter formula:

Perimeter = 12 × a

The dodecagon area formula is more complicated, so we accompany it with an approximation:

Area = 3 × (2 + √3) ≈ 11.196 × a²

More information on how to calculate the area of a dodecagon awaits you in Omni's
dodecagon area calculator!

The formulas for the radius of the circumcircle R and the radius of the incircle r are the following:

R = ½ a × (√6 + √2) ≈ 1.932 × a

r = ½ a × (2 + √3) ≈ 1.866 × a

## What are the angles in a dodecagon?

The interior angle α in a dodecagon has 150°. As there are twelve identical interior angles, the sum of all the interior angles of a dodecagon equals 1800°.

We can also easily deduce that the exterior angle, β, has 30°, because it satisfies α + β = 180°. Note that these angles remain the same no matter the side length of the regular dodecagon!

## How to use this dodecagon calculator?

To use Omni's dodecagon calculator, simply input whatever data you have: side length, perimeter, area, incircle or circumcircle radius, or any of the diagonals... Our tool will immediately compute all the remaining values, and you'll have the complete set of parameters of the dodecagon.

Tip. Once you're done with dodecagons, you can play with other regular polygons: just hit the advanced mode button beneath the dodecagon calculator and set the number of sides to whatever value you like!

See how fascinating dodecagons are? There are plenty of mathematical challenges around regular polygons! To help you satisfy your curiosity, we've built a whole bunch of polygon-related tools. We highly recommend you visit a few of them:

## FAQ

### How many sides does a dodecagon have?

A dodecagon is a polygon with twelve sides as well as twelve vertices and twelve internal angles. This fact is encoded in the word itself, as dodeka is Greek for twelve.

### How many diagonals does a dodecagon have?

A dodecagon has 54 diagonals. To get this result:

1. Recall that a polygon with n sides has n(n - 3)/2 diagonals.
2. Plug in n = 12 and compute 12 × (12 - 3) / 2 = 54. This is our result.
3. If you forget the formula, compute in how many ways we can choose two out of twelve vertices: it is 12 × 11 / 2 = 66. However, this result includes both sides and diagonals.
4. Subtract twelve sides: 66 - 12 = 54, and that's it.

### What is the area of a dodecagon with a 10 cm side?

The answer is approximately 1120 cm². To get this result, use the formula area ≈ 11.196 × a². We plug in a = 10 and obtain area ≈ 11.196 × 100 ≈ 1119.6 ≈ 1120. This is an approximation of the precise formula that reads area = 2.5 × a² × √(5 + 2 × √5).

### How do I find the sum of the interior angles of a dodecagon?

The interior angles of a dodecagon sum up to 1800°. To arrive at this result, recall that the formula for the sum of internal angles in a polygon with n sides reads n × 180° - 360°. Plugging in n = 12, we obtain 12 × 180° - 360° = 2160° - 360° = 1800°. This is our answer.

Anna Szczepanek, PhD Parameters
Side (a)
in
Perimeter
in
Area
in²
in
in
α
150
deg
β
30
deg
Diagonal across 2 sides
in
Diagonal across 3 sides
in
Diagonal across 4 sides
in
Diagonal across 5 sides
in
Diagonal across 6 sides
in
For other regular polygons, click the advanced mode button below.
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