The area of an octagon is the region covered by the octagon’s boundaries in two-dimensional (2D) space. There are a few ways to calculate the area of polygons. One way is to use the length of the apothem and the length of one of its sides. However, it is also possible to calculate the area of an octagon simply by using the length of one of its sides.

Here, we will learn about the two main formulas that we can use to calculate the area of octagons. In addition, we will solve some exercises in which we will apply these two formulas.

## Formula to find the area of an octagon

We can find the area of any polygon by dividing it into congruent triangles and calculating the area of each triangle. In the case of octagons, we can divide the figure into eight congruent triangles as in the following figure:

We know that the area of any triangle is equal to $latex A=\frac{1}{2} bh$, where *b* is the length of the base and *h* is the length of the height. In the figure above, we see that the height of the triangle is equal to the apothem and the base of the triangle is equal to one of the sides of the octagon, so we have $latex A = \frac{1}{2} sa$.

Also, we know that we have eight of these triangles in the octagon, so the formula for the area of an octagon is as follows:

$latex A=4sa$ |

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

### Area of an octagon without using the apothem

We can also calculate the area of a regular octagon without using the length of its apothem. This requires that we find the length of the apothem in terms of the length of the sides.

Using trigonometry and simplifying, we can find the following formula for the area of an octagon in terms of its sides:

$latex A=2(1+\sqrt{2}){{s}^2}$ |

## Area of an octagon – Examples with answers

The following examples can be used to practice using the formulas for the area of an octagon. Each example has its respective solution, but it is recommended that you try to solve the exercises yourself before looking at the answer.

**EXAMPLE 1**

An octagon has sides of length 5 m and an apothem of length 6.04 m. What is its area?

##### Solution

We have the following values:

- Sides, $latex s=5$ m
- Apothem, $latex a=6.04$ m

Depending on this, we can use the first formula:

$latex A=4sa$

$latex A=4(5)(6.04)$

$latex A=120.8$

The area of the octagon is 120.8 m².

**EXAMPLE 2**

What is the area of an octagon that has sides of length 6 m and an apothem of 7.24 m?

##### Solution

We can obtain the following information:

- Sides, $latex s = 6$ m
- Apothem, $latex a=7.24$ m

We substitute these values in the formula for the area:

$latex A=4sa$

$latex A=4(6)(7.24)$

$latex A=173.76$

The area of the octagon is 173.76 m².

**EXAMPLE 3**

What is the area of an octagon that has sides of length 11 m and an apothem of length 13.28 m?

##### Solution

We have the following information:

- Sides, $latex s=11$ m
- Apothem, $latex a=13.28$ m

We use this data in the first formula:

$latex A=4sa$

$latex A=4(11)(13.28)$

$latex A=584.32$

The area of the octagon is 584.32 m².

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**EXAMPLE 4**

What is the area of an octagon that has sides of length 8 m?

##### Solution

Here, we only have the length of one side of the octagon, so we have to use the second formula with $latex s = 8$:

$latex A=2(1+\sqrt{2}){{s}^2}$

$latex A=2(1+\sqrt{2}){{(8)}^2}$

$latex A=2(1+\sqrt{2})(64)$

$latex A=309.02$

The area of the octagon is 309.02 m².

**EXAMPLE 5**

What is the area of an octagon that has sides of length 10 m?

##### Solution

We use the second formula for the area with $latex s = 10$:

$latex A=2(1+\sqrt{2}){{s}^2}$

$latex A=2(1+\sqrt{2}){{(10)}^2}$

$latex A=2(1+\sqrt{2})(100)$

$latex A=482.84$

The area of the octagon is 482.84 m².

## Area of an octagon – Practice problems

Solve the following problems to practice using the formulas for the area of an octagon seen above. If you need help with these problems, you can look at the solved examples above.

## See also

Interested in learning more about octagons? Take a look at these pages:

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