# Dodecagon

A dodecagon is a polygon with 12 sides and 12 vertices.

## What is a dodecagon

A dodecagon is a 12-sided two-dimensional plane figure. The term dodecagon is derived from Greek, where "dodeka" indicates 12 (do = two, deka = 10) and gon means angle or corner. The figure below shows some dodecagon examples:

## Dodecagon types

Dodecagons are categorized as regular or irregular and concave or convex.

### Regular dodecagon

A regular dodecagon is one in which all sides and interior angles have the same measure. A regular dodecagon has 12 equal sides and interior angles. It is also always convex (described below). The figure below shows a regular dodecagon example.

### Irregular dodecagon

An irregular dodecagon is any dodecagon that has at least one side (and therefore angle) that has a different measure. The figure below shows an irregular dodecagon example.

### Convex dodecagon

A convex dodecagon is a dodecagon in which all interior angles measure less than 180°. As a result, it is not possible to draw any line segment between the vertices of a convex dodecagon such that the line segment passes outside of the dodecagon. The figure below shows a convex dodecagon example (one that is not a regular dodecagon).

### Concave dodecagon

A concave dodecagon is one in which at least one interior angle measures greater than 180°. As a result, it is possible to draw a line segment between at least one pair of vertices such that the line segment lies outside of the boundaries of the dodecagon. The figure below shows a concave dodecagon example.

## Dodecagon properties

Below are some properties of all dodecagons as well as properties of a regular dodecagon.

### Properties of all dodecagons

All dodecagons have the following properties:

- All dodecagons have 12 sides, 12 vertices, and 12 angles.
- All dodecagons have 54 diagonals.
- All dodecagons have interior angles that sum to 1800°.
- All dodecagons have exterior angles that sum to 360°.
- All dodecagons have diagonals that divide it into 10 triangles drawn from its vertices.

### Properties of regular dodecagons

- The interior angles of a regular dodecagon each measure 150°.
- The exterior angles of a regular dodecagon each measure 30°.
- The 10 triangles formed by the diagonals of a regular dodecagon are congruent.

## Parts of a dodecagon

A dodecagon is made up of sides, angles, and vertices; it has 12 of each. It also contains 54 diagonals.

### Dodecagon angles

Dodecagons can be broken into a series of triangles by diagonals drawn from its vertices. This series of triangles can be used to find the sum of the interior angles of the dodecagon.

Diagonals are drawn from vertex A in the convex dodecagon below, forming 10 triangles. Similarly, 10 triangles can also be drawn in a concave dodecagon. Since the sum of the angles in a triangle is 180°, the sum of the interior angles of a dodecagon is 10 × 180° = 1800°.

A regular dodecagon has equal interior angle measures. Since 1800°/12 = 150°, each interior angle in a regular dodecagon has a measure of 150°. Also, each exterior angle has a measure of 30° since the exterior angle and interior angle form a straight angle.

### Dodecagon diagonals

A dodecagon has 54 diagonals. A diagonal is a line segment drawn from a vertex of a polygon to a non-adjacent vertex. The figure below shows the diagonals of a regular dodecagon drawn for just one of the vertices. The diagonals are drawn in the same way for the rest of the vertices of the dodecagon.

## Dodecagon area

There are a few formulas for finding the area of a regular dodecagon. Refer to the following figure of a regular dodecagon for each of the formulas.

- Side - the side of a dodecagon is one of its 12 edges.
- Apothem - the apothem of a dodecagon is the perpendicular distance from the center of the dodecagon to one of its edges.
- Circumradius - the circumradius of a dodecagon is the radius of the circle within which the dodecagon is inscribed. This distance can be measured as the distance from the center of the dodecagon to one of its vertices.

### Area using side length

The area of a regular dodecagon with side length a is:

### Area using apothem

The area of a regular dodecagon with apothem length r is:

### Area using circumradius

The area of a regular dodecagon with circumradius R is:

## Dodecagon perimeter

The perimeter of a regular dodecagon can be found using the circumradius or apothem.

### Perimeter using circumradius

The perimeter of a regular dodecagon with circumradius R is:

### Perimeter using apothem

The perimeter of a regular dodecagon with apothem r is: