Composite Volume: Problems, Comparisons and Mistakes | Grade 5 Math | LevelKite

You already know how to split a solid into two box-shaped parts and add the space they fill. Now you will put that skill to work. You will solve story problems, compare two boxes, and check work for mistakes.

Here are the words you need. Volume is the amount of space a solid fills. You measure it by counting unit cubes, which are cubes with edges $1$ unit long. That is why volume is given in cubic units, such as cubic feet. A right rectangular prism is a box shape. Each of its six faces is a rectangle. A composite solid is a solid made by joining simpler solids. In this lesson, each composite solid is made of two right rectangular prisms.

Now think about the cubes. When two prisms are joined and share no space, no cube is inside both of them. Each unit cube sits in exactly one prism. So if you count the cubes in each prism and then add, you count every cube once.

Key concept

Count every cube exactly once

Volume is a count of unit cubes. In a solid made of two prisms that share no space, each cube sits in exactly one prism. So you find each prism's volume from its own length, width and height, and then you add the two volumes. Every problem and every check in this lesson comes back to one question: was each cube counted once, and only once?

\text{total volume} = \text{volume of prism 1} + \text{volume of prism 2}

A story problem hides the two prisms inside its words. Your first job is to find them.

Read slowly. For each prism, write down its three edge lengths. An edge length is how long one edge of the prism is. You need three of them: the length, the width and the height. In a story, the height may be called the depth.

Then follow the same plan each time:

Find the volume of the first prism. Multiply its own length, width and height.
Find the volume of the second prism the same way, with its own edge lengths.
Add the two volumes, and name the cubic unit in your answer.

Worked example

A pool with two depths

A swimming pool has two parts joined end to end. Each part is shaped like a right rectangular prism. The shallow part is $9$ meters long, $5$ meters wide and $1$ meter deep. The deep part is $7$ meters long, $5$ meters wide and $3$ meters deep. How many cubic meters of water fill the whole pool?

List the edge lengths of each prism so that no number gets mixed up. Shallow part: 9 meters long, 5 meters wide, 1 meter deep. Deep part: 7 meters long, 5 meters wide, 3 meters deep. The depth is the height of each prism.
Find the volume of the shallow part. Multiply its own three edge lengths, because that counts the 1-meter cubes that fill this prism.
$9 \times 5 \times 1 = 45$
Find the volume of the deep part the same way. Use only the deep part's edge lengths, because you are counting the cubes that fill the deep part.
$7 \times 5 \times 3 = 105$
Add the two volumes. The parts are joined end to end and share no space, so each cubic meter of water is in exactly one part.
$45 + 105 = 150$

Answer

The whole pool is filled by $150$ cubic meters of water.

Next, compare two prisms. To compare volumes is to say which prism fills more space, and by how much.

Do not judge by one edge. A prism that is $9$ by $2$ by $3$ has a volume of $9 \times 2 \times 3 = 54$ cubic units. A prism that is $5$ by $5$ by $3$ has a volume of $5 \times 5 \times 3 = 75$ cubic units. The first prism has the longest edge, but the second prism fills more space. It fills $75 - 54 = 21$ more cubic units. Volume depends on all three edge lengths together.

To see how one edge changes the volume, picture a prism as a stack of equal layers. Length times width tells you how many cubes are in one layer. The height tells you how many layers there are. Here is a prism that is $4$ long, $3$ wide and $2$ tall, written as layers.

\underbrace{4 \times 3}_{\text{cubes in one layer}} \times \underbrace{2}_{\text{layers}} = 24

Keep the 4 by 3 layer of 12 cubes and change only the height

Height (number of layers)	Number sentence	Compared with height 2
$2$	$4 \times 3 \times 2 = 24$	Starting prism
$3$	$4 \times 3 \times 3 = 36$	One more layer, so $12$ more cubes
$4$	$4 \times 3 \times 4 = 48$	$2$ times the layers, so $2$ times the volume
$6$	$4 \times 3 \times 6 = 72$	$3$ times the layers, so $3$ times the volume

Read down the table. One more layer means $12$ more cubes, so $24 + 12 = 36$ . A height that is $2$ times as great gives $2$ times as many layers, so the volume is $2$ times as great: $2 \times 24 = 48$ .

The length and the width work the same way. You can turn a prism on its side so that any edge becomes its height. So the pattern is this: when one edge length is multiplied by a number and the other two edge lengths stay the same, the volume is multiplied by that same number.

Worked example

Which box holds more?

Box A is $8$ centimeters (cm) long, $5$ cm wide and $2$ cm tall. Box B is $8$ cm long, $5$ cm wide and $6$ cm tall. Which box has the greater volume? How many times as great is it?

Look for what is the same and what is different. Both boxes are 8 cm long and 5 cm wide, so one layer of each box has the same number of cubes: 40. Only the height is different.
$8 \times 5 = 40$
Find the volume of Box A. It is 2 cm tall, so it has 2 layers of 40 cubes.
$8 \times 5 \times 2 = 80$
Find the volume of Box B. It is 6 cm tall, so it has 6 layers of 40 cubes.
$8 \times 5 \times 6 = 240$
Compare the volumes. Box B is 3 times as tall as Box A, because 6 is 3 times 2. That gives it 3 times as many layers, so its volume is 3 times as great.
$240 = 3 \times 80$

Answer

Box B has the greater volume. Its volume is $240$ cubic centimeters, which is $3$ times the $80$ cubic centimeters of Box A.

Now you are the checker. When you check work on a solid made of two prisms, ask three questions:

Is any part counted twice?
Does each edge length belong to the prism it is used for?
Were the two volumes added?

Try the questions on a solid made of two blocks. Each block is a right rectangular prism. The bottom block lies flat on a table. It is $10$ cm long, $5$ cm wide and $2$ cm tall. The top block stands on one end of the bottom block. It is $3$ cm long, $5$ cm wide and $6$ cm tall. From the table to the highest point is $2 + 6 = 8$ cm.

Correct work looks like this. Bottom block: $10 \times 5 \times 2 = 100$ . Top block: $3 \times 5 \times 6 = 90$ . Total: $100 + 90 = 190$ cubic centimeters. The table shows three ways this work can go wrong.

Three mistakes on the two-block solid

Mistake	What was written	Why it fails	Correct work
A shared part is counted twice	Bottom $10 \times 5 \times 2 = 100$ , top $3 \times 5 \times 8 = 120$ , total $220$	The $8$ cm includes $2$ cm of the bottom block. Those $3 \times 5 \times 2 = 30$ cubes are counted twice.	Top $3 \times 5 \times 6 = 90$ , total $100 + 90 = 190$
An edge length belongs to the other prism	Bottom $10 \times 5 \times 6 = 300$	The $6$ cm is the height of the top block. The bottom block is $2$ cm tall.	Bottom $10 \times 5 \times 2 = 100$
The volumes are multiplied	$100 \times 90 = 9{,}000$	That would mean $100$ groups of $90$ cubes. The solid is one group of $100$ and one group of $90$ .	$100 + 90 = 190$

Worked example

Find and fix the mistake

A wooden stand has a low step and a high step, side by side. Each step is a right rectangular prism that sits on the floor. Both steps are $4$ feet wide. The low step is $3$ feet long and $1$ foot tall. The high step is $5$ feet long and $2$ feet tall. The whole stand is $8$ feet long. Mia writes: bottom layer $8 \times 4 \times 1 = 32$ , high step $5 \times 4 \times 2 = 40$ , total $32 + 40 = 72$ cubic feet. Find and fix her mistake.

Ask the first checking question: is any part counted twice? Mia's multiplying is right, so look at her two prisms. Her bottom layer is the lowest 1 foot of the whole stand, so it includes the bottom half of the high step. Her high step starts at the floor, so it includes that same bottom half. Her two prisms share space.
Find the volume of the shared part. It is the bottom half of the high step: 5 feet long, 4 feet wide and 1 foot tall. Mia counted these cubes twice.
$5 \times 4 \times 1 = 20$
Fix the work with two prisms that share no space. Keep the high step. Beside it, use only the low step, which is 3 feet long, 4 feet wide and 1 foot tall.
$3 \times 4 \times 1 = 12$
Add the volume of the low step to the volume of the high step. Now every cube is counted once.
$12 + 40 = 52$
Check another way. Mia's total should be too big by exactly the shared part, so take the shared part away once.
$72 - 20 = 52$

Answer

Mia counted the bottom half of the high step twice. The correct volume of the stand is $52$ cubic feet, not $72$ cubic feet.

Here is the idea to keep. Volume counts unit cubes, and every cube must be counted exactly once.

For a solid made of two prisms, find each prism's volume from its own three edge lengths, then add.
To compare two prisms, compare their volumes. Changing one edge changes the number of layers.
To check work, ask three questions. Is any part counted twice? Does each edge length belong to its prism? Were the volumes added?