The surface area of a triangular prism represents the two-dimensional surface covered by the prism. On the other hand, the volume is a measure of the three-dimensional space occupied by the prism. **We can calculate the surface area of a triangular prism, by adding the areas of all its faces, and we can calculate the volume using the formula V=½bah, where, b is the base of the triangular face, a is the height of the triangular face and h is the height of the prism.**

In this article, we will explore the formulas that we can use to calculate the surface area and volume of a triangular prism. Then we’ll use these formulas to solve some practice problems.

##### GEOMETRY

**Relevant for**…

Learning to calculate the surface area and volume of triangular prisms.

##### GEOMETRY

**Relevant for**…

Learning to calculate the surface area and volume of triangular prisms.

## How to find the surface area of a triangular prism?

We can calculate the surface area of a triangular prism by adding the areas of the faces of the prism. A triangular prism has two equal triangular faces and three rectangular faces that may or may not be equal depending on the type of triangle we have at the bases.

The area of each triangular face is equal to $latex \frac{1}{2}ab$, where *a* is the height of the triangular base and *b* is the length of its base. Since both triangular bases are equal, the area of both triangular faces is $latex ab$.

In addition, we have three lateral rectangular faces. The area of each rectangular face is equal to the height of the prism multiplied by the three sides of the triangular base.

This means that 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, the formula for the surface area of a triangular prism is:

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

If the bases of the prism are equilateral triangles, we know that the three sides of the triangle are equal, so the three areas of the lateral faces are equal.

## How to find the volume of a triangular prism?

To calculate the volume of a triangular prism, we have to multiply the area of the triangular bases by the height of the prism. This means that we have to find the area of one of the triangular faces, and then multiply it by the height of the prism.

Since the area of any triangle can be found by dividing the product of its height and base by 2, the formula for the volume of a triangular prism is:

$latex V=\frac{1}{2}b\times a\times h$ |

where

*b*is the base of the triangular face*a*is the height of the triangular face*h*is the height of the prism

## Surface area and volume of triangular prisms – Practice problems

The formulas for the surface area and volume of a triangular prism are used to solve the following examples. Try to solve the problems yourself before looking at the solution.

**EXAMPLE 1**

Find the surface area of a triangular prism that has a height of 6 inches and its triangular base has sides with lengths of 5 inches, 6 inches, 5 inches, and a height of 4 inches.

##### Solution

We have the following lengths:

- Height of prism, $latex h=6$
- Side 1, $latex b_{1}=5$
- Side 2, $latex b_{2}=6$
- Side 3, $latex b_{3}=5$
- Height of triangle, $latex a=4$

Using the formula for the surface area, we have:

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

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

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

$latex A_{s}=120$

The surface area is equal to 120 in².

**EXAMPLE **2

**EXAMPLE**

Calculate the volume of a triangular prism that has a height of 5 feet and its triangular base has a height of 3 feet and a base of 4 feet.

##### Solution

We have the following:

- Height of prism, $latex h=5$
- Height of triangle, $latex a=3$
- Base of triangle, $latex b=4$

Applying the volume formula with these data, we have:

$latex V=\frac{1}{2}bah$

$latex V=\frac{1}{2}(4)(3)(5)$

$latex V=30$

The volume is equal to 30 ft³.

**EXAMPLE **3

**EXAMPLE**

Find the surface area of a triangular prism that has a height of 10 feet and its triangular base has sides with lengths of 13 feet, 10 feet, 13 feet, and a height of 12 feet.

##### Solution

We can observe the following lengths:

- Height of prism, $latex h=10$
- Side 1, $latex b_{1}=13$
- Side 2, $latex b_{2}=10$
- Side 3, $latex b_{3}=13$
- Height of triangle, $latex a=12$

We use these values in the formula for the surface area and 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 equal to 480 ft².

**EXAMPLE **4

**EXAMPLE**

Find the volume of a triangular prism that has a height of 6 inches and its triangular base has a height of 5 inches and a base of 6 inches.

##### Solution

We have the following:

- Height of prism, $latex h=6$
- Height of triangle, $latex a=5$
- Base of triangle, $latex b=6$

Using the volume formula with the given lengths, we have:

$latex V=\frac{1}{2}bah$

$latex V=\frac{1}{2}(6)(5)(6)$

$latex V=90$

The volume is equal to 90 in³.

**EXAMPLE **5

**EXAMPLE**

What is the surface area of a triangular prism that has a height of 5 inches, an equilateral base with sides of 6 inches, and a height of 5.2 inches?

##### Solution

We have the following

- Height of prism, $latex h=5$
- Side, $latex b=6$
- Height of triangle, $latex a=5.2$

Since the bases are equilateral triangles, all three side faces will have the same area. Therefore, we have:

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

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

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

$latex A_{s}=31.2+90$

$latex A_{s}=121.2$

The surface area is equal to 121.2 in².

**EXAMPLE **6

**EXAMPLE**

What is the volume of a prism that has a height of 8 yards and its triangular base has a height of 6 yards and a base of 7 yards?

##### Solution

We have the following information:

- Height of prism, $latex h=8$
- Height of triangle, $latex a=6$
- Base of triangle, $latex b=7$

Using this in the volume formula, we have:

$latex V=\frac{1}{2}bah$

$latex V=\frac{1}{2}(7)(6)(8)$

$latex V=168$

The volume is equal to 168 yd³.

**EXAMPLE **7

**EXAMPLE**

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

##### Solution

We have the following dimensions:

- Height of prism, $latex h=8$
- Side, $latex b=9$
- Height of triangle, $latex a=7.8$

Using the formula for the surface area with the given lengths, we have:

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

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

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

$latex A_{s}=70.2+216$

$latex A_{s}=286.2$

The surface area is equal to 286.2 ft².

**EXAMPLE **8

**EXAMPLE**

Find the volume of a prism that has a height of 11 feet and its triangular base has a height of 5 feet and a base of 4 feet.

##### Solution

We have the following:

- Height of prism, $latex h=11$
- Height of triangle, $latex a=5$
- Base of triangle, $latex b=4$

Using the volume formula with these lengths, we have:

$latex V=\frac{1}{2}bah$

$latex V=\frac{1}{2}(4)(5)(11)$

$latex V=110$

The volume is equal to 110 ft³.

## Surface area and volume of triangular prisms – Practice problems

Use everything you have learned about the surface area and volume of a triangular prism to solve the following problems.

## See also

Interested in learning more about the area and volume of geometric figures? Take a look at these pages:

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