Chapter 11 · Volume

Volume Word Problems and Missing Edge Lengths

Solve word problems about the volume of box shapes by choosing the formula that fits. Then find a missing edge length from the volume and check it.

15 min read

Planters, closets and shipping boxes are often shaped like a right rectangular prism. That is a box shape with six flat faces that are all rectangles, so its sides stand straight up from its bottom. Every box in this lesson has that shape.

The volume of a box is the amount of space inside it. You measure volume in cubic units. One cubic foot is the space inside a cube that is $1$ foot long, $1$ foot wide and $1$ foot tall.

You already know two formulas for volume. A formula is a rule written with letters. In this lesson you will use them in word problems. A word problem is a short story with a math question in it. These stories do not tell you which formula to use. You read what each story gives you, and then you choose.

A story can give you these facts:

Edge lengths. An edge is a line where two faces meet. Its edge length is how long that line is. A box has three edge lengths: its length $l$ , its width $w$ and its height $h$ .
The area of the base. The base is the face the box sits on. Its area, $B$ , is the amount of flat space it covers, in square units such as square feet.
The volume, $V$ .

Read what the story gives you, then choose the formula

What the story gives you	Formula that fits	With numbers
Three edge lengths: a crate is $5$ feet long, $3$ feet wide and $2$ feet tall.	$V = l \times w \times h$	$V = 5 \times 3 \times 2 = 30$ , so the volume is $30$ cubic feet.
The area of the base and the height: a crate's base covers $15$ square feet, and the crate is $2$ feet tall.	$V = B \times h$	$V = 15 \times 2 = 30$ , so the volume is $30$ cubic feet.

Both rows describe the same crate. In the second story, the length and the width are already multiplied for you: $5 \times 3 = 15$ . That is why the two formulas give the same volume. $B$ is the area of the base, and the area of the base is $l \times w$ .

Before you use either formula, make sure of two things:

The shape is a right rectangular prism.
The units match. Every edge length uses the same unit, such as feet. Then the base area is in square feet and the volume is in cubic feet.

Read the question, too. In this lesson, questions such as How much space is inside? and How much does it take to fill it? ask for the volume.

Worked example

A word problem: choose the formula

A raised garden bed is shaped like a right rectangular prism. Its base covers $18$ square feet of ground. The bed is $2$ feet tall. How many cubic feet of soil does it take to fill the bed to the top?

Sort the facts in the story. The base covers 18 square feet, so the area of the base is B = 18. The bed is 2 feet tall, so the height is h = 2. Soil that fills the bed to the top takes up all the space inside, so the question asks for the volume, V.
$B = 18 \qquad h = 2 \qquad V = \text{?}$
Choose the formula. The story gives the area of the base and the height. It does not give a length and a width. So the formula that fits is volume equals base area times height.
$V = B \times h$
Put the numbers in place of the letters, then multiply. This works because each 1 foot of height holds a layer of 18 cubic feet of soil, and the bed is 2 feet tall. Volume is measured in cubic units, so the unit is cubic feet.
$V = 18 \times 2 = 36$

Answer

It takes $36$ cubic feet of soil to fill the garden bed to the top.

Now turn the question around. Some stories tell you the volume and leave out one edge length. That is a missing edge length. You can still find it. Here is why.

Picture a box packed full with $120$ small cubes. Each cube is $1$ centimeter (cm) long, $1$ cm wide and $1$ cm tall, so each cube fills $1$ cubic centimeter. The volume of the box is $120$ cubic centimeters. The bottom of the box is $6$ cm long and $4$ cm wide. How tall is the box?

Look at one flat layer of cubes on the bottom. It has $6$ rows of $4$ cubes, which is $6 \times 4 = 24$ cubes. Each cube covers $1$ square centimeter of the bottom, so $24$ is also the area of the base in square centimeters. Every layer above it has $24$ cubes too.

So ask: how many layers of $24$ cubes make $120$ cubes? That is a division question, and $120 \div 24 = 5$ . The box has $5$ layers. Each layer is $1$ cm tall, so the box is $5$ cm tall.

The cube box in words, in the formula and in numbers

In the cube box	In the formula	With numbers
One layer has $24$ cubes.	Area of the base, $B$ : $24$ square centimeters	$6 \times 4 = 24$
The whole box has $120$ cubes.	Volume, $V$ : $120$ cubic centimeters	$V = 120$
The box has $5$ layers.	Height, $h$ : $5$ centimeters	$120 \div 24 = 5$

Key concept

Multiply to find volume, divide to find an edge

The volume of a right rectangular prism comes from multiplying its length, its width and its height. The base area $B$ is the length times the width.

So when you know every number but one, you can find the one that is left:

Volume missing? Multiply what you know.
Edge length missing? Divide the volume by what you know: the base area, or the two known edge lengths multiplied together. Dividing by that number undoes multiplying by it.

This works for a missing length, width or height, because you can multiply in any order. The units must match, such as feet, square feet and cubic feet.

V = l \times w \times h = B \times h

Worked example

A missing height from the base area

A closet is shaped like a right rectangular prism. It has $96$ cubic feet of space inside. Its floor has an area of $12$ square feet. How high is the ceiling above the floor?

Sort the facts. The space inside is the volume, so V = 96. The floor is the base, so B = 12. The distance from the floor up to the ceiling is the height, h. That is the missing edge length.
$V = 96 \qquad B = 12 \qquad h = \text{?}$
The story gives the area of the base, so use the formula with base area. Put in the two numbers you know. The height is the one number still missing from the multiplication.
$V = B \times h \text{, so } 96 = 12 \times h$
Ask: 12 times what number makes 96? Divide 96 by 12 to find out, because dividing by 12 undoes multiplying by 12. A height is a length, so its unit is feet, not square feet or cubic feet.
$h = 96 \div 12 = 8$
Check by multiplying back. The base area times the height you found should give the volume from the story. It does, so 8 feet is correct.
$12 \times 8 = 96$

Answer

The ceiling is $8$ feet above the floor, so the closet is $8$ feet tall.

A found edge length is easy to check. In the closet problem, you multiplied to see if you got the volume back. You can do the same with edge lengths.

The check: multiply the two edge lengths from the story by the edge length you found. If the answer is the volume from the story, your edge length is correct. If it is a different number, look for a mistake.

Try it on the cube box: $6 \times 4 \times 5 = 120$ . That is $5$ layers of $24$ cubes, which makes all $120$ cubes. So $5$ cm is correct.

This check needs all three edge lengths. When a story gives only the base area, check with $B \times h$ , as you did for the closet.

Worked example

A missing width, with a check

A shipping box is a right rectangular prism with a volume of $240$ cubic inches. The box is $8$ inches long and $5$ inches tall. How wide is the box?

Sort the facts. The volume is 240 cubic inches, the length is 8 inches and the height is 5 inches. The width, w, is the missing edge length. The story gives edge lengths, so write the formula with three edge lengths and put in the numbers you know.
$V = l \times w \times h \text{, so } 240 = 8 \times w \times 5$
Multiply the two edge lengths you know. You can multiply in any order, so join the 8 and the 5 first. Now the width is the only number missing.
$8 \times 5 = 40 \text{, so } 240 = 40 \times w$
Divide the volume by 40, because dividing by 40 undoes multiplying by 40. A width is a length, so its unit is inches.
$w = 240 \div 40 = 6$
Check with all three edge lengths: the 8 and the 5 from the story, and the 6 you found. They multiply to 240, the volume from the story, so the width is correct.
$8 \times 6 \times 5 = 240$

Answer

The shipping box is $6$ inches wide.

Here is the big idea to take with you. The volume of a right rectangular prism is made by multiplying, so each problem in this lesson was a multiplication with one number missing.

Volume missing? Multiply. Use $V = l \times w \times h$ when the story gives three edge lengths. Use $V = B \times h$ when it gives the area of the base and the height.
Edge length missing? Divide the volume by what you know.
Then check. Multiply all three edge lengths, or the base area and the height. You should get the volume from the story.

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