Math Core

Not every solid is a simple box. Steps, buildings, and many packages are made of two or more boxes stuck together. You can find their volume by breaking them into rectangular prisms and adding.

Break it apart

Look at this solid. It is shaped like the letter L.

An L-shaped solid. The colored line splits it into a tall box and a short box.

There is no single length, width, and height for the whole shape, so $l \times w \times h$ won't work all at once. Instead, split it into two boxes that don't overlap.

Tall box (left): 2 cm long, 3 cm wide, 4 cm tall.
Short box (right): The whole bottom is 5 cm long and the tall box uses 2 cm of it. So the short box is $5 - 2 = 3$ cm long. It is 3 cm wide and 2 cm tall.

Now find each volume and add:

\begin{aligned} \text{Tall box: } & 2 \times 3 \times 4 = 24 \\ \text{Short box: } & 3 \times 3 \times 2 = 18 \\ \text{Total: } & 24 + 18 = 42 \end{aligned}

The volume of the solid is 42 cubic centimeters.

Volume of a combined solid

Split the solid into rectangular prisms that don't overlap.
Find the volume of each prism.
Add the volumes.

Another way to split

You can often split a solid in more than one way. The answer must come out the same, because the solid holds the same amount of space.

Worked example: Split it the other way

Split the same L-shaped solid with a flat cut instead.

Bottom box: the full 5 cm long, 3 cm wide, and 2 cm tall. Volume: $5 \times 3 \times 2 = 30$ .
Top box: 2 cm long and 3 cm wide. The whole solid is 4 cm tall, and the bottom box uses 2 cm, so this box is $4 - 2 = 2$ cm tall. Volume: $2 \times 3 \times 2 = 12$ .

Total: $30 + 12 = 42$ cubic cm. Same answer!

Worked example: A stacked cake

A cake has two layers. The bottom layer is 6 inches long, 6 inches wide, and 2 inches tall. The top layer is 3 inches long, 3 inches wide, and 2 inches tall. What is the total volume of the cake?

\begin{aligned} \text{Bottom: } & 6 \times 6 \times 2 = 72 \\ \text{Top: } & 3 \times 3 \times 2 = 18 \\ \text{Total: } & 72 + 18 = 90 \end{aligned}

The cake has a volume of 90 cubic inches.

Worked example: A pool with a deep end

A pool has a shallow part and a deep part, side by side. The shallow part is 6 m long, 5 m wide, and 1 m deep. The deep part is 4 m long, 5 m wide, and 3 m deep. How much water fills the pool?

\begin{aligned} \text{Shallow part: } & 6 \times 5 \times 1 = 30 \\ \text{Deep part: } & 4 \times 5 \times 3 = 60 \\ \text{Total: } & 30 + 60 = 90 \end{aligned}

The pool holds 90 cubic meters of water.

Common mistake

Watch the side lengths of each piece. In the L-shaped solid, the short box is not 5 cm long. The 5 cm is the whole bottom edge, and part of it belongs to the tall box. Subtract to find the missing length.

Tip

Check your work by splitting the solid a second way. If both totals match, you can trust your answer.

Practice

Practice 1

A solid is made of two boxes that don't overlap. One box has a volume of 30 cubic cm. The other has a volume of 45 cubic cm. What is the volume of the solid, in cubic cm?

Type a number.

Practice 2

A small box sits on top of a bigger box. The bigger box is 5 ft long, 4 ft wide, and 2 ft tall. The small box is 2 ft long, 2 ft wide, and 3 ft tall. What is the total volume, in cubic feet?

Type a number.

Practice 3

A solid is made of box A, which is 4 cm by 2 cm by 3 cm, and box B, which is 5 cm by 2 cm by 1 cm. Which expression gives its volume?

Practice 4

Find the volume of this solid, in cubic inches.

The tall part is 2 in long and 3 in tall. The whole solid is 6 in long and 2 in wide. The short part is 1 in tall.

Type a number.

Practice 5

A set of steps is made of three boxes side by side. Each box is 4 ft wide and 1 ft deep. The boxes are 1 ft, 2 ft, and 3 ft tall. What is the total volume of the steps, in cubic feet?

Type a number.

Practice 6

An L-shaped solid is 9 ft long across the bottom and 4 ft wide. Its tall part is 3 ft long and 6 ft tall. The rest of the solid is 2 ft tall. What is its volume, in cubic feet?

Type a number.

Practice 7

A pool has a shallow part that is 10 m long, 8 m wide, and 1 m deep. Next to it is a deep part that is 5 m long, 8 m wide, and 3 m deep. How many cubic meters of water fill the whole pool?

Type a number.