# Area and Volume of a Rectangular Prism – Formulas and Examples

The surface area of a rectangular prism represents the two-dimensional surface area occupied 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 rectangular prism using the formula A=2(bl+lh+hb) and we can calculate its volume using the formula V=lbh, where l is the width, b is 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 rectangular 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 rectangular prism.

See examples

##### GEOMETRY

Relevant for

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

See examples

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

To calculate the surface area of a rectangular prism, we have to add the areas of the six faces of the prism. Generally, a rectangular prism has three dimensions with different lengths, as shown in the diagram below.

Noting that all the faces of the prism are rectangular, we can calculate its area by multiplying the two dimensions of each face. Therefore, the formula for the surface area of this rectangular prism is:

where,

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

This formula is derived by considering that parallel faces in a rectangular prism have the same area.

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

We can calculate the volume of a rectangular prism by multiplying the lengths of its three dimensions. That is, we have to multiply the lengths of the base, width, and height of the prism.

Therefore, the formula for the volume of a rectangular prism is:

where,

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

Volume is represented in cubic units.

## Surface area and volume of rectangular prisms – Examples with answers

The formulas for the surface area and volume of a rectangular 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 rectangular prism that has a base of 5 inches, a width of 4 inches, and a height of 4 inches.

We have the following:

• Base, $latex b=5$
• Width, $latex l=4$
• Height, $latex h=4$

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

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

$latex A_{s}=2((5)(4)+(4)(4)+(4)(5))$

$latex A_{s}=2(20+16+20)$

$latex A_{s}=2(56)$

$latex A_{s}=112$

The surface area is equal to 112 in².

### EXAMPLE 2

Find the volume of a rectangular prism that has a base of 5 feet, a width of 4 feet, and a height of 4 feet.

We have the following dimensions:

• Base, $latex b=5$
• Width, $latex l=4$
• Height, $latex h=4$

Applying the formula for the volume with these dimensions, we have:

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

$latex V=5 \times 4 \times 4$

$latex V=80$

The volume is equal to 80 ft³.

### EXAMPLE 3

Find the surface area of a rectangular prism that has a base of 7 yards, a width of 6 yards, and a height of 8 yards.

We have the following information:

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

Using the formula for the surface area with these values, 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 yd².

### EXAMPLE 4

Find the volume of a rectangular prism that has a base of 8 inches, a width of 6 inches, and a height of 7 inches.

We have the following lengths:

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

Substituting these values into the formula for the volume, we have:

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

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

$latex V=336$

The volume is equal to 336 in³.

### EXAMPLE 5

What is the surface area of a rectangular prism that has a base of 8 inches, a height of 12 inches, and a width of 11 inches?

We have the following information:

• Base, $latex b=8$
• Width, $latex l=11$
• Height, $latex h=12$

Applying the surface area formula, we have:

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

$$A_{s}=2((8)(11)+(11)(12)+(12)(8))$$

$latex A_{s}=2(88+132+96)$

$latex A_{s}=2(316)$

$latex A_{s}=632$

The surface area is equal to 632 in².

### EXAMPLE 6

Find the volume of a rectangular prism that has a base of 10 feet, a width of 11 feet, and a height of 12 feet.

We can observe the following lengths:

• Base, $latex b=10$
• Width, $latex l=11$
• Height, $latex h=12$

When we use this information in the formula for the volume, we have:

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

$latex V=10 \times 11 \times 12$

$latex V=1320$

The volume is equal to 1320 ft³.

### EXAMPLE 7

Find the height of a rectangular prism that has a surface area of 148 yd² if its base is 6 yards long and its width is 4 yards long.

We have the following:

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

In this case, we know the surface area, and we need to find the length of the height. Therefore, we use the formula for the surface area 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 yards long.

### EXAMPLE 8

Find the height of a rectangular prism that has a base of 5 inches, a width of 3 inches, and a volume of 90 in³.

We have the following information:

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

In this case, we know the volume of the prism, and we need to find the length of the height. Therefore, we have 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 inches long.

### EXAMPLE 9

Determine the height of a rectangular prism that has a surface area of 340 in², a width of 5 inches, and a base of 8 inches.

We have the following:

• Base, $latex b=8$
• Width, $latex l=5$
• Superficial area, $latex A=340$

We use the formula for the surface area with these data and solve for h:

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

$latex 340=2((8)(5)+(5)h+(8)(h))$

$latex 340=2(40+13h)$

$latex 170=40+13h)$

$latex 13h=170-40$

$latex 13h=130$

$latex h=10$

The height is equal to 10 inches.

### EXAMPLE 10

Find the height of a rectangular prism that has a volume of 693 ft³, a base of 11 feet, and a width of 9 feet.

We have the following:

• Base, $latex b=11$
• Width, $latex l=9$
• Volume, $latex V=693$

Let’s use the volume formula and solve for h:

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

$latex 693=11 \times 9 \times h$

$latex 693=99h$

$latex h=7$

The length of the height is 7 ft.

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

Solve the following problems by applying everything you have learned about the surface area and the volume of a rectangular prism.

#### Find the height of a rectangular prism that has a volume of 240 yd3, a width of 5 yards, and a base of 6 yards.  