Area and Volume of a Prism – Formulas and Examples

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

In this article, we will learn all about the surface area and volume of a prism. We will explore their formulas and use them to solve some practice problems.

GEOMETRY

Relevant for

Learning to calculate the surface area and volume of a prism.

See examples

GEOMETRY

Relevant for

Learning to calculate the surface area and volume of a prism.

See examples

How to find the surface area of a prism?

The surface area of any prism can be calculated by adding the areas of all the faces of the prism. Depending on the type of prism we have, we will have a different number of faces.

In a prism, we have two bases that have the same dimensions and the same area. Also, we have several lateral faces that may or may not have the same area depending on whether the bases are regular.

For example, in the triangular prism that we can see in the following diagram, we can find the area of the bases with the formula $latex \frac{1}{2}ab$, where a is the height of the triangular base and b is the length of its base. This means that the area of both triangular faces is $latex ab$.

Also, the area of the rectangular lateral faces is equal to the height of the prism multiplied by each side of the triangular base. That is, we have the areas $latex b_{1}h$, $latex b_{2}h$ and $latex b_{3}h$. Therefore, the surface area of the triangular prism is:

In the case of a rectangular prism, we have six rectangular faces. Generally, these prisms have their three dimensions with different lengths, as shown in the diagram below.

Therefore, considering that the parallel faces in a rectangular prism have the same area, we can obtain the following formula for its surface area:

where

• b is the base of the prism
• l is the width of the prism
• h is the height of the prism

These ideas can be applied to calculate the surface area of any prism.

How to find the volume of a prism?

The volume of any prism can be calculated by multiplying the area of its base by the height of the prism. Therefore, we can use the following formula:

The area of the base of the prism will depend on the type of prism we have. For example, in a triangular prism, we can calculate the area of its base by multiplying one-half the length of the base by the length of the height. Then, we have the following formula:

where

• b is the base of the triangle
• a is the height of the triangle
• h is the height of the prism

In the case of a rectangular prism, we can calculate the area of its base by multiplying the length of the width by the length of the base. Therefore, we have the following:

where

• l is the width of the prism
• b is the base of the prism
• h is the height of the prism

We can use these ideas to calculate the volume of any prism.

Surface area and volume of prisms – Examples with answers

In the following examples, we have to find the surface area and volume of various prisms. Try to solve the problems yourself before looking at the solution.

EXAMPLE 1

What is the surface area of a triangular prism that has a height of 10 inches and its triangular base has sides with the lengths 13, 10, 13 inches, and a height of 12 inches?

We have the following information:

• 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, $latex a=12$

With this, we can find the area of both triangular bases and the area of the three lateral rectangular faces. Therefore, 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 equal to 480 in².

EXAMPLE 2

Find 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.

We have the following:

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

To find the volume, we have to multiply the area of the base by the height of the prism. Therefore, we have:

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

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

$latex V=168$

The volume is 168 equal to yd³.

EXAMPLE 3

What is the surface area of a rectangular prism that has a base of 7 inches, a width of 6 inches, and a height of 8 inches?

We have the following:

• Base, $latex b=7$
• Width, $latex l=6$
• Height, $latex h=8$

We have to find the areas of the six rectangular faces, considering that the parallel faces have the same area. Therefore, we have:

$latex A_{s}=2(bl+lh+hb)$

$latex A_{s}=2((7)(6)+(6)(8)+(8)(7))$

$latex A_{s}=2(42+48+56)$

$latex A_{s}=2(146)$

$latex A_{s}=292$

The surface area is equal to 292 in².

EXAMPLE 4

What is the volume of a rectangular prism that has a base of 8 feet, a width of 6 feet, and a height of 7 feet?

We have the following information:

• Base, $latex b=8$
• Width, $latex l=6$
• Height, $latex h=7$

To find the volume of the rectangular prism, we simply have to multiply the lengths of its three dimensions:

$latex V=b \times l \times h$

$latex V=8 \times 6 \times 7$

$latex V=336$

The volume is equal to 336 ft³.

EXAMPLE 5

Determine 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.

We have the following lengths:

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

Since the bases of the prism are equilateral triangles, their sides have the same length. Therefore, we calculate its surface area as follows:

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

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

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

$latex A_{s}=31.2+90$

$latex A_{s}=121.2$

The surface area is equal to 121.2 in².

EXAMPLE 6

Find 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.

We have the following lengths:

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

We find the volume of the prism by multiplying the area of the triangular base by the height of the prism:

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

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

$latex V=168$

The volume is equal to 168 yd³.

EXAMPLE 7

What is the height of a rectangular prism with a surface area of 148 in² if its base is 6 inches long and its width is 4 inches long?

We have the following:

• Base, $latex b=6$
• Width, $latex l=4$
• Surface area, $latex A=148$

In this case, we have the surface area of the rectangular prism, and we need to find the length of the height. Therefore, we use the surface area formula and solve for h:

$latex A_{s}=2(bl+lh+hb)$

$latex 148=2((6)(4)+(4)h+(6)(h))$

$latex 148=2(24+10h)$

$latex 74=24+10h)$

$latex 10h=74-24$

$latex 10h=50$

$latex h=5$

The height is 5 inches long.

EXAMPLE 8

What is the height of a rectangular prism that has a base of 5 feet, a width of 3 feet, and a volume of 90 ft³?

We have the following:

• Base, $latex b=5$
• Width, $latex l=3$
• Volume, $latex V=90$

In this case, we are going to use the formula for the volume and solve for h:

$latex V=b \times l \times h$

$latex 90=5 \times 3 \times h$

$latex 90=15h$

$latex h=6$

The height is 6 feet long.

EXAMPLE 9

Find the surface area of a hexagonal prism that has a height of 5 inches and a hexagonal base with sides that have a length of 3 inches.

We have the following:

• Sides of hexagon, $latex a=3$
• Height of prism, $latex h=5$

A hexagonal prism has two hexagonal bases and six rectangular lateral faces. The area of the six rectangular faces is equal to 6ah and the area of the hexagonal faces is equal to $latex 3\sqrt{3}{{a}^2}$. Therefore, we have:

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

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

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

$latex A_{s}=46.77+90$

$latex A_{s}=136.77$

The surface area is equal to 136.77 in².

EXAMPLE 10

Find the volume of a hexagonal prism that has sides with a length of 4 feet and a height of 6 feet.

We have the following lengths:

• Sides of hexagon, $latex a=4$
• Height, $latex h=6$

The volume of the hexagonal prism is equal to the area of the hexagonal base multiplied by the height of the prism:

$latex V=\frac{3\sqrt{3}}{2}{{a}^2}h$

$latex V=\frac{3\sqrt{3}}{2}{{(4)}^2}(6)$

$latex V=\frac{3\sqrt{3}}{2}(16)(6)$

$latex V=249.4$

The volume is equal to 249.4 ft³.

Surface area and volume of prisms – Practice problems

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