The surface area of a triangular prism is the total area covered by the prism. Surface area is a two-dimensional measure, so we can use m², cm² or others. To calculate the surface area of any 3D figure, we have to add the measures of the areas of all the faces of the figure. A triangular prism has two equal triangular faces and three rectangular faces.

Here, we will learn about the formula that we can use to calculate the surface area of a rectangular prism. Also, we will use this formula to solve some practice problems.

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

The formula for the surface area of a triangular prism is obtained by adding the expressions for the areas of all the faces of the prism. In a triangular prism, we have two equal triangular faces and three rectangular faces that may or may not be the same.

Each triangular face has an area of $latex \frac{1}{2} ab$, where *a* is the length of the height of the triangular base and *b* is the length of its base. This means that the area of both triangular faces is $latex ab$.

The area of each rectangular face is equal to the height of the prism multiplied by the three sides of the triangular base. That is, we have the areas $latex b_{1} h$, $latex b_{2}h$ and $latex b_{3} h$, where, $latex b_{1}, ~ b_{2}, ~ b_{3}$ are the lengths of the sides of the triangular base and *h* is the length of the height of the prism.

Therefore, by adding all these areas, we have:

$latex A_{s}=ab+b_{1}h+b_{2}h+b_{3}h$ |

In the case that the base is an equilateral triangle, we know that the three sides of the triangle are equal, so the three areas of the lateral faces are equal.

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

The following examples are used to practice using the formula for the surface area of triangular prisms. Try to solve the problems yourself before looking at the solution.

**EXAMPLE 1**

A triangular prism has a height of 6 m and its triangular base has sides of length 5 m, 6 m, 5 m, and a height of 4 m. What is its surface area?

##### Solution

We can obtain the following information:

- Prism height, $latex h=6$
- Side 1, $latex b_{1}= 5$
- Side 2, $latex b_{2} = 6$
- Side 3, $latex b_{3}=5$
- Triangle height, $latex a=4$

We use the formula for surface area and substitute the given values:

$latex A_{s}=ab+b_{1}h+b_{2}h+b_{3}h$

$$A_{s}=(4)(6)+(5)(6)+(6)(6)+(5)(6)$$

$latex A_{s}=24+30+36+30$

$latex A_{s}=120$

The surface area is 120 m².

**EXAMPLE 2**

A triangular prism has a height of 10 m and its triangular base has sides of length 13 m, 10 m, 13 m, and a height of 12 m. What is its surface area?

##### Solution

We have the following values:

- Prism height, $latex h=10$
- Side 1, $latex b_{1}=13$
- Side 2, $latex b_{2}=10$
- Side 3, $latex b_{3}=13$
- Triangle height, $latex a=12$

We use these values in the formula for surface area and we have:

$latex A_{s}=ab+b_{1}h+b_{2}h+b_{3}h$

$$A_{s}=(12)(10)+(13)(10)+(10)(10)+(13)(10)$$

$latex A_{s}=120+130+100+130$

$latex A_{s}=480$

The surface area is 480 m².

**EXAMPLE 3**

We have a triangular prism that has an equilateral base with sides 6 m and a height of 5.2. If the height of the prism is 5 m, what is its surface area?

##### Solution

In this case, we have an equilateral triangle, so we know that the sides of the triangle are equal. Therefore, we have:

- Prism height, $latex h=5$
- Side, $latex b=6$
- Triangle height, $latex a=5.2$

We use the formula of the surface area with this information and we combine the three areas of the lateral sides since they are equal:

$latex A_{s}=ab+bh+bh+bh$

$latex A_{s}=ab+3bh$

$latex A_{s}=(5.2)(6)+3(6)(5)$

$latex A_{s}=31.2+90$

$latex A_{s}=121.2$

The surface area is 121.2 m².

**EXAMPLE 4**

A prism has a base that is an equilateral triangle with sides of length 9 m and a height of 7.8 m. If the height of the prism is 8 m, what is its surface area?

##### Solution

We have the following information:

- Prism height, $latex h=8$
- Side, $latex b=9$
- Triangle height, $latex a = 7.8$

We use the formula for surface area and substitute the given values:

$latex A_{s}=ab+b_{1}h+b_{2}h+b_{3}h$

$latex A_{s}=ab+3bh$

$latex A_{s}=(7.8)(9)+3(9)(8)$

$latex A_{s}=70.2+216$

$latex A_{s}=286.2$

The surface area is 286.2 m².

## Surface area of a triangular prism – Practice problems

Practice using the surface area of triangular prisms to solve the following problems. Select an answer and check it to make sure you selected the correct one. If you need help with this, you can look at the solved examples above.

## See also

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

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