The surface area of a hexagonal prism represents the total area occupied by the prism. We use square units to measure surface area since it is a two-dimensional measure. The surface area of a hexagonal prism can be calculated by adding the areas of all its faces. These prisms are composed of two congruent hexagonal faces and six rectangular lateral faces that are also congruent.

Here, we will learn the formula for the surface area of these prisms, which is obtained by adding the expressions of the areas of each face. Also, we will use this formula to solve some practice problems.

## Formula to find the surface area of a hexagonal prism

Hexagonal prisms have two hexagonal faces that are parallel and congruent. These prisms also have six lateral rectangular faces that are also congruent. To find its surface area, we have to add the areas of all these faces. Therefore, we start by finding expressions for a hexagonal face and a rectangular face.

The area of a regular hexagon can be found using the formula: $latex A=\frac{3 \sqrt{3}}{2}{{a}^2}$, where, *a* is the length of one of the sides of the hexagon. This means that the area of both hexagonal faces is $latex A=3 \sqrt{3}{{a}^2}$.

The area of one of the rectangular faces of the prism is equal to *ah*, where *a* is the length of one of the sides of the hexagon and *h* is the height of the prism. Therefore, the area of the six rectangular faces equals 6*ah*.

By adding the expressions obtained for the areas, we have:

$latex A_{S}=3\sqrt{3}{{a}^2}+6ah$ |

## Surface area of a hexagonal prism – Examples with answers

The following examples are solved using the formula for the surface area of hexagonal prisms. Each example has its respective solution, but it is recommended that you try to solve the exercises yourself before looking at the result.

**EXAMPLE 1**

A hexagonal prism has a height of 5 m and a hexagonal base with sides of length 3 m. What is its surface area?

##### Solution

From the question, we have the following:

- Hexagon sides, $latex a=3$
- Prism height, $latex h=5$

We substitute the values given in the formula for surface area:

$latex A_{S}=3\sqrt{3}{{a}^2}+6ah$

$latex A_{S}=3\sqrt{3}{{(3)}^2}+6(3)(5)$

$latex A_{S}=3\sqrt{3}(9)+90$

$latex A_{S}=46.77+90$

$latex A_{S}=136.77$

The surface area is 136.77 m².

**EXAMPLE 2**

What is the surface area of a hexagonal prism that has a hexagonal base with sides of length 4 m and a height of 6 m?

##### Solution

We have the following values:

- Hexagon sides, $latex a=4$
- Prism height, $latex h=6$

Using the formula for the surface area with these values, we have:

$latex A_{S}=3\sqrt{3}{{a}^2}+6ah$

$latex A_{S}=3\sqrt{3}{{(4)}^2}+6(4)(6)$

$latex A_{S}=3\sqrt{3}(16)+144$

$latex A_{S}=83.14+90$

$latex A_{S}=173.14$

The surface area is 173.14 m².

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**EXAMPLE 3**

We have a prism with a height of 10 m and a hexagonal base with sides with a length of 5 m. What is its surface area?

##### Solution

We acknowledge the following:

- Hexagon sides, $latex a=5$
- Prism height, $latex h=10$

Using the formula for surface area with these values, we have:

$latex A_{S}=3\sqrt{3}{{a}^2}+6ah$

$latex A_{S}=3\sqrt{3}{{(5)}^2}+6(5)(10)$

$latex A_{S}=3\sqrt{3}(25)+300$

$latex A_{S}=129.9+300$

$latex A_{S}=429.9$

The surface area is 429.9 m².

**EXAMPLE 4**

What is the height of a prism that has a surface area of 226.77 m² and a hexagonal base with sides of length 3 m?

##### Solution

We have the following values:

- Hexagon sides, $latex a=3$
- Surface area, $latex A = 226.77$

In this case, we have the surface area and we want to find the length of the height of the prism. Therefore, we use the values given in the formula and solve for *h*:

$latex A_{S}=3\sqrt{3}{{a}^2}+6ah$

$latex 226.77=3\sqrt{3}{{(3)}^2}+6(3)h$

$latex 226.77=46.77+18h$

$latex 18h=226.77-46.77$

$latex 18h=180$

$latex h=10$

The length of the height is equal to 10 m.

**EXAMPLE 5**

If a prism has a surface area of 542.6 m² and the hexagonal base has sides of length 6 m, what is the length of its height?

##### Solution

We have the following:

- Hexagon sides, $latex a=6$
- Surface area, $latex A=542.6$

We use the formula for surface area with the given values and solve for *h*:

$latex A_{S}=3\sqrt{3}{{a}^2}+6ah$

$latex 542.6=3\sqrt{3}{{(6)}^2}+6(6)h$

$latex 542.6=254.6+36h$

$latex 36h=542.6-254.6$

$latex 36h=288$

$latex h=8$

The length of the height is equal to 8 m.

## Surface area of a hexagonal prism – Practice problems

Test the use of the formula for the surface area of hexagonal prisms and 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 hexagonal prisms? Take a look at these pages:

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