# Dodecagon: Definition, Angles, Area, and Properties

## TL;DR
A dodecagon is a polygon with 12 sides, 12 vertices, and 12 angles. Its interior angles always add to 1800°, a regular dodecagon has each interior angle equal to 150°, and it carries 54 diagonals. This article covers the definition, the four types, the angle, perimeter, and area formulas with their derivations, and the mistakes students make with the 12-sided shape.

## What Is A Dodecagon?
A **dodecagon** is a closed two-dimensional polygon with **12 straight sides, 12 vertices, and 12 interior angles**. The name comes from the Greek _dodeka_, meaning twelve, and _gon_, meaning sides or angles. Like every polygon, a dodecagon is defined by its side count alone — the sides can be equal or unequal, and the shape can bulge outward or cave inward.

When all 12 sides and all 12 angles are equal, it is a **regular dodecagon**, the version you meet most often. A dodecagon belongs to the same family as the [pentagon](/content/math/geometry/pentagon-shape/index.html) and [hexagon](/content/math/geometry/hexagon/index.html) — just with more sides.

## The Four Types Of Dodecagon
A dodecagon can be sorted by whether its sides and angles are equal, and by whether it bulges out or caves in.

- **Regular dodecagon** — all 12 sides equal and all 12 interior angles equal (150° each).
- **Irregular dodecagon** — sides and angles are not all equal; only the side count of 12 is fixed.
- **Convex dodecagon** — every interior angle is less than 180°, so no vertex points inward.
- **Concave dodecagon** — at least one interior angle is greater than 180° (a reflex angle), so one vertex caves inward.

## Angles Of A Dodecagon
**Sum of interior angles.** Any polygon splits from one vertex into (n − 2) triangles, each worth 180°. The [interior angles](/content/math/geometry/interior-angles/index.html) of a polygon therefore total:

\[\text{Sum of interior angles} = (n - 2) \times 180°\]

For a dodecagon, n = 12:

\[(12 - 2) \times 180° = 10 \times 180° = 1800°\]

A dodecagon splits into exactly **10 triangles**, which is why the total is 10 lots of 180°.

**Each interior angle of a regular dodecagon.** Share the 1800° total across 12 equal corners:

\[\text{Each interior angle} = \frac{1800°}{12} = 150°\]

**Each exterior angle of a regular dodecagon.** Interior and exterior angles at a vertex make a straight line:

\[\text{Each exterior angle} = 180° - 150° = 30°\]

The exterior angles of any polygon add to 360°, which checks the answer: 360° ÷ 12 = 30°.

**Variable glossary:** n is the number of sides (12 here); (n − 2) is the number of triangles; 180° is one triangle's angle sum.

## Perimeter Of A Dodecagon
The **perimeter** is the total distance around the shape — the sum of all 12 side lengths. For a regular dodecagon with side length s, every side is the same, so:

\[P = 12s\]

If a regular dodecagon has sides of 4 cm, its perimeter is 12 × 4 cm = 48 cm. For an irregular dodecagon you simply add the 12 individual side lengths; there is no shortcut.

## Area Of A Dodecagon
For a **regular dodecagon** with side length s, the area formula is:

\[A = 3(2 + \sqrt{3})s^{2}\]

**Where this comes from:** a regular dodecagon can be cut into 12 identical isosceles triangles meeting at the center. Each triangle has area \(\frac{1}{2} s^{2} \cot(15°)\) and twelve of them yield the clean \(3(2 + \sqrt{3}) s^{2}\).

**Variable glossary:** A is the area, s is the length of one side, and 3(2 + √3) ≈ 11.196 is the constant that bundles the 12 triangles together.

| Property                       | Formula                     | Regular value (n = 12) |
|-------------------------------|-----------------------------|--------------------------|
| Sum of interior angles        | (n − 2) × 180°             | 1800°                   |
| Each interior angle           | 1800° ÷ 12                 | 150°                    |
| Each exterior angle           | 180° − 150°                | 30°                     |
| Number of diagonals           | n(n − 3) ÷ 2               | 54                       |
| Perimeter                     | 12s                        | 12s                      |
| Area                          | 3(2 + √3) s²               | ≈ 11.196 s²              |

## How Many Diagonals Does A Dodecagon Have?
A **diagonal** joins two non-adjacent vertices. From each of the 12 vertices you can draw a diagonal to 9 others (you skip the vertex itself and its 2 neighbors). That counts every diagonal twice, so divide by 2:

\[\text{Number of diagonals} = \frac{n(n-3)}{2} = \frac{12 \times 9}{2} = 54\]

## Examples Of The Dodecagon
### Example 1
**Find the sum of the interior angles of a dodecagon.**

Sum = (n − 2) × 180°  
Sum = (12 − 2) × 180°  
**Final answer:** 1800°

### Example 2
**A student claims each interior angle of a regular dodecagon is 1800° ÷ 10 = 180°. Find the correct value.**
Take the wrong path first, because this is a common error.  
Wrong attempt: 1800° ÷ 10 = 180°  
Correct method: **Final answer:** 150°

### Example 3
**A regular dodecagon has a side length of 5 cm. Find its perimeter.**  
**Final answer:** 60 cm

### Example 4
**Find each exterior angle of a regular dodecagon.**  
**Final answer:** 30°

### Example 5
**Find the number of diagonals in a dodecagon.**  
**Final answer:** 54

### Example 6
**A regular dodecagon has a side of 4 cm. Find its area.**  
**Final answer:** ≈ 179.1 cm²

## Why Twelve Sides Shows Up So Often
Twelve is a number that divides cleanly — by 2, 3, 4, and 6. A regular dodecagon's angles and layout enable practical applications in design and engineering.

## Tripping Points To Avoid
**Mistake 1:** Dividing 1800° by 10 instead of 12.  
**Mistake 2:** Applying 150° to an irregular dodecagon.  
**Mistake 3:** Miscounting diagonals as n(n − 1) ÷ 2.

## Conclusion
- A **dodecagon has 12 sides, 12 vertices, and 12 angles**.
- Its **interior angles always add to 1800°**.
- A **regular dodecagon** has each interior angle of 150° and each exterior angle of 30°.
- It carries **54 diagonals**.
- The regular area is **3(2 + √3) s²** and the perimeter is **12s**.