# Decagon

A decagon is a polygon with 10 sides and 10 vertices.

## Decagon meaning

A decagon gets its name from the Ancient Greek word "deka-" meaning "ten" and "gonia" meaning corner or angle. Thus, a decagon is a 10-sided closed-figure with 10 corners and angles. There are various types of decagons that can look quite different.

## What does a decagon look like

Decagons only need to be closed, two-dimensional shapes with 10 sides, angles, and vertices. For this reason, they can take on many different shapes. The figure below shows a few decagon examples:

## Decagon types

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

### Regular decagon

A regular decagon is one in which all sides and interior angles have the same measure. A regular decagon has 10 equal sides and interior angles. It is also always convex. The figure below shows a regular decagon example:

### Irregular decagon

An irregular decagon is any decagon that has at least one side and angle that has a different measure. The figure below shows an irregular decagon example:

### Convex decagon

A convex decagon is a decagon whose interior angles all measure less than 180°. It is not possible to draw any line segment between the vertices of a convex decagon such that the line segment passes outside of the boundaries of the decagon.

Note that a regular decagon is always convex, but a convex decagon is not always a regular decagon. The figure below shows a convex decagon example (one that is not a regular decagon).

### Concave decagon

A concave decagon is one in which at least one of the interior angles has a measure greater than 180°. For a decagon to be concave, it must be possible to draw a line segment between at least one pair of its vertices such that the line segment lies outside the boundaries of the decagon.

### Complex decagon

A complex decagon is a decagon that is self-intersecting that has additional interior spaces. All of the decagons described above are referred to as simple decagons, since they do not meet this criteria. The figure below shows a complex decagon example:

## Decagon properties

Below are some properties of all decagons as well as properties of a regular decagon.

### Properties of all decagons

All decagons have the following properties:

- All decagons have 10 sides, 10 vertices, and 10 angles.
- All decagons have 35 diagonals.
- All decagons have 10 interior angles that sum to 1440°.
- All decagons have 10 exterior angles that sum to 360°.
- All decagons have diagonals that divide it into 8 triangles drawn from its vertices.

### Properties of regular decagons

Regular decagons additionally have the following properties:

- All regular decagon interior angles measure 144° each.
- All regular decagon exterior angles measure 36° each.
- The 8 triangles formed by the diagonals of a regular decagon are congruent.

## Angles of a decagon

Decagons 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 decagon.

Diagonals are drawn from vertex A in the convex decagon below, forming 8 triangles. Similarly, 8 triangles can also be drawn in a concave decagon. Since the sum of the angles of a triangle is 180°, the sum of the interior angles of a decagon is 8 × 180° = 1440°.

A regular decagon has equal interior angle measures. Since 1440°/10 = 144°, each interior angle in a regular decagon has a measure of 144°. Also, each exterior angle has a measure of 36°.

## Decagon area

There are a few formulas for finding the area of a regular decagon. Refer to the figure below for all of the following formulas.

- side (a) - the side of a decagon is one of its 10 edges.
- apothem (r) - the apothem of a decagon is the perpendicular distance from the center of the decagon to one of its edges.
- circumradius (R) - the circumradius of a decagon is the radius of the circle within which the decagon is inscribed. The distance can be measured as the distance from the center of the decagon to one of its vertices.

### Area of a decagon using side length

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

### Area of a decagon using apothem

### Area of a decagon using circumradius

## Decagon perimeter

The perimeter of a regular decagon with side length a is:

The perimeter of an irregular decagon is the sum of all of its edges.

### Did you know?

A regular decagon can be inscribed in a circle. Each vertex on the decagon lies on the circle. Also, the circle and decagon share the same center.