Octagons are polygons that have eight sides and eight interior angles. The total sum of the interior angles in an octagon is equal to 1080°. We can define some types of octagons depending on the characteristics that we consider. For example, based on the lengths of the sides, we can have regular octagons and irregular octagons.

Here, we will look at a definition of octagons and learn about some of the most important types of octagons. Also, we will learn about the fundamental characteristics of these geometric figures. In addition, we will review the most important formulas for solving octagon problems and apply them to solve some problems.

## Definition of an octagon

We can define an octagon as a polygon with eight sides and eight interior angles. The sides of the octagon meet each other end to end to form a figure in the two-dimensional (2D) plane. Taking different characteristics, we can identify the following types of octagons:

- Regular and irregular
- Convex and concave

### Irregular and regular octagons

A regular octagon is a figure that has all its sides equal and all its angles equal in measure. On the other hand, a regular octagon is a figure that has sides of different lengths, angles of different measures, or both at the same time.

### Convex and concave octagons

A convex octagon is a figure that has all its vertices pointing outward. A concave octagon is a figure that has at least one vertex pointing inward.

## Fundamental characteristics of an octagon

The following are the fundamental characteristics of octagons:

- The sum of all the interior angles is 1080°.
- Each internal angle in a regular octagon is equal to 135°.
- A regular octagon has all eight sides with the same length.
- A regular octagon has all eight angles with the same measure.
- The sum of the exterior angles is equal to 360°.
- Each external angle in a regular octagon is equal to 45°.

## Important octagon formulas

The following are the most commonly used formulas to solve octagon problems.

### Formula for the perimeter of an octagon

The perimeter is equal to the sum of the lengths of all the sides. In a regular octagon, all the sides are equal, so we have:

$latex p=6s$ |

where *s* is the length of the sides of the octagon.

### Formula for the area of a regular octagon

The formula for the area of a regular octagon is as follows:

$latex A=4 sa$ |

where *s* is the length of one of the sides and *a* is the length of the apothem.

### Formula of the apothem of an octagon

We can calculate the length of the apothem of a regular octagon with the following formula:

$latex a= \frac{s}{2\tan(22.5)}$ |

where *s* is the length of one side of the octagon.

## Examples of octagon problems

### EXAMPLE 1

- An octagon has sides of length 12 m. What is its perimeter?

**Solution:** We have the length $latex s = 12$. Therefore, we use the perimeter formula with this value:

$latex p=8s$

$latex p=8(12)$

$latex p=96$

The perimeter is 96 m.

### EXAMPLE 2

- What is the area of an octagon that has sides of length 7 m and an apothem of length 8.45 m?

**Solution:** We have the values $latex s = 7$ and $latex a=8.45$, so we replace them in the area formula:

$latex A=4sa$

$latex A= 4(7)(8.45)$

$latex A=236.6$

The area of the octagon is 236.6 m².

### EXAMPLE 3

- An octagon has sides that are 15 m long. What is its apothem?

**Solution:** We use the value $latex s = 15$ in the apothem formula:

$latex a= \frac{s}{2\tan(22.5)}$

$latex a= \frac{15}{2\tan(22.5)}$

$latex a= \frac{15}{0.828}$

$latex a=18.11$

The length of the apothem is 18.11 m.

## Octagon – Practice problems

## See also

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