The area of a hexagon is the confined space within the sides of the hexagon, that is, it is the two-dimensional region covered by the figure. Fundamentally, we can find the area of hexagons using the length of their sides and the length of their apothem. However, there is also a formula that allows us to find its area simply by using the length of one side of the hexagon.

Here, we will learn about the two main formulas that we can use to find the area of hexagons. In addition, we will apply these formulas to solve some practice problems.

## Formula to find the area of a hexagon

A formula for finding the area of a regular hexagon is as follows:

$latex A=3sa$ |

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

We can derive this formula by dividing the hexagon into six congruent triangles as in the following figure.

We know that the area of a triangle is $latex A = \frac{1}{2} bh$, where *b* is the length of the base and *h* is the length of the height. In this case, the base of each triangle is equal to one side of the hexagon and the height is equal to the apothem, so the area of each triangle is equal to $latex A=\frac{1}{2}sa$.

Since we have 6 of these triangles in the hexagon, we multiply the area obtained by 6 to obtain $latex A= 3sa$, which is the area of the hexagon.

### Area of the hexagon without using the apothem

Alternatively, it is also possible to find the area of a hexagon without using the length of the apothem. For this, we have to find the length of the apothem in terms of the length of one of the sides of the hexagon.

We know that the triangles formed in a hexagon are equilateral, so the height of each triangle is given by $latex h = \frac{\sqrt{3}}{2} s$, where *s* is the length of one of the sides of the hexagon. Therefore, by substitutting this value for the value of *a*, we have:

$latex A=3s(\frac{\sqrt{3}}{2}s)$ $latex A=\frac{3\sqrt{3}}{2}{{s}^2}$ |

## Area of a hexagon – Examples with answers

The following examples use the formulas for the area given above. Each exercise has its respective solution, where you can look at the process and reasoning used.

**EXAMPLE 1**

What is the area of a hexagon that has sides of length 4 m and an apothem of length 3.46 m?

##### Solution

We have the following information:

- Sides, $latex s=4$ m
- Apothem, $latex a=3.46$ m

Depending on this, we can use the first formula:

$latex A=3sa$

$latex A=3(4)(3.46)$

$latex A=41.52$

The area of the hexagon is 41.52 m².

**EXAMPLE 2**

A hexagon has sides of length 6 cm and apothem of length 5.2 cm. What is its area?

##### Solution

We have the following values:

- Sides, $latex s=6$ cm
- Apothem, $latex a=5.2$ cm

We use the first formula with these values:

$latex A=3sa$

$latex A=3(6)(5.2)$

$latex A=93.6$

The area of the hexagon is 93.6 cm².

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**EXAMPLE 3**

What is the area of a hexagon that has sides of length 7 m and an apothem of length 6.06 m?

##### Solution

We use the following information:

- Sides, $latex s=7$ m
- Apothem, $latex a=6.06$ m

We substitute this in the first formula:

$latex A=3sa$

$latex A=3(7)(6.06)$

$latex A=127.26$

The area of the hexagon is 127.26 m².

**EXAMPLE 4**

What is the area of a hexagon that has sides of length 5 m?

##### Solution

In this case, we only have the length of one side of the hexagon, so we use the second formula with $latex s = 5$:

$latex A=\frac{3\sqrt{3}}{2}{{s}^2}$

$latex A=\frac{3\sqrt{3}}{2}{{(5)}^2}$

$latex A=\frac{3\sqrt{3}}{2}(25)$

$latex A=64.96$

The area of the hexagon is 64.96 m².

**EXAMPLE 5**

A pentagon has sides that are 8 m long. What is its area?

##### Solution

Again, we have to use the second formula since we only have the length of one side of the hexagon. Thus, we substitute $latex s=6$ in the formula:

$latex A=\frac{3\sqrt{3}}{2}{{s}^2}$

$latex A=\frac{3\sqrt{3}}{2}{{(8)}^2}$

$latex A=\frac{3\sqrt{3}}{2}(64)$

$latex A=166.28$

The area of the hexagon is 166.28 m².

## Area of a hexagon – Practice problems

Practice using the formulas for the area of a hexagon to solve the following problems. If you need help with this, you can look at the solved examples above.

## See also

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

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