**What is an Octagon?**

**Regular Octagon:** A regular octagon is a plane or flat shape with 8 equal straight sides. The shape is closed, meaning all of the lines are connected.

A “Stop” sign is a regular octagon.

A regular octagon has:

- 8 equal sides
- 8 interior angles of 135 degrees each
- 8 exterior angle of 45 degrees

**Note: **It’s possible to have an irregular octagon with 8 unequal sides and angles, but this page is devoted to calculating the area of a regular octagon.

**How to Calculate the Area of a Regular Octagon**

There is more than one way to figure out the area of an octagon.

- A formula can be used to calculate the area of a regular octagon.
- There are different ways to deconstruct the octagon into different shapes, as if you were cutting it apart. You then calculate the areas of the different shapes and add them together.

**Method 1: Formula for Calculating the Area of a Regular Octagon**

Area of an octagon: 2 x a^{2 }x (1 + )

= 1.414

**Solution 1**

a = one short side of the octagon = 9

2 x 9^{2} x (1 + 1.414)

2 x 81 x (1 + 1.414) = 391.068

**Solution 2**

a = one short side of the octagon = 17

2 x 17^{2} x (1 + 1.414)

2 x 289 x (1 + 1.414) = 1,395.292

**Method 2: Dividing a Regular Octagon into 8 Triangles**

Divide the octagon into 8 triangles. Each triangle has 2 sides of equal length.

**Solution**

In this example, b = 24 and h = 29.

Area = ½ (b x h) (base x height)

½ of (24 x 29)

24 x 29 = 696 / 2 = 348

348 x 8 = 2,784 = area of octagon

**Method 3: Dividing an Octagon into 1 Square, 4 Rectangles and 4 Isosceles Right Triangles**

Divide the octagon into:

1 square: area = a^{2}

4 rectangles: area of 1 rectangle = a x b

4 triangles: Area of 1 triangle = ½ (b x h)

Add the area totals of 1 square, 4 rectangles and 4 triangles together for the area of the octagon.

There are additional methods of determining the area of an octagon. Use the method that works best for you.

**Fun Fact About Octagons**

Can you imagine a house with 8 sides? Between the 1850s and the early 1900s, there was a brief period when 8-sided or octagon homes became somewhat popular. A few are still standing today and much appreciated.

An amateur architect, Orson Squire Fowler, believed the natural geometry of the octagon offered a lot of advantages. He said these homes would receive more natural light, use space efficiently and heat and cool easily. Victorian builders were already familiar with building 135 degree corners and had no problem building these unusual homes. (Octagonal house photo by Sanfranman59)