Grade 5 · 5.MD.C.5 · 5.MD.C.5C

Volume of a solid made of boxes

A step stool, a medal podium and a planter box are not single boxes. How do you find how much space each one takes up, or holds?

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Volume of a boxGrade 5

Where you are 9 in Volume and capacity

Measurement12

Volume and capacity9

▴ 4 above Volume with unit cubes Volume of a box Surface area, and how it compares with volume Volume of a box with fraction edge lengths

Volume of a solid made of boxes

When you reach for it

Seojun is building a wooden step stool. Pooja is measuring the podium where the medal winners will stand. Wapi is filling a planter box with soil. Each of them wants a volume, and none of the three things is a single box.

Volume of a box gives the volume of one box: length times width times height. This lesson handles solids that are made of several boxes. There are two moves. You can split the solid into boxes, find each volume, and add. Or you can see the solid as one big box with a box-shaped part taken away, and subtract. You reach for this whenever a solid has steps, notches or hollows, but every face is still a flat rectangle meeting the others at square corners. It is the same thinking as Area of composite figures and shaded regions, one dimension up.

The lesson runs over three days. Day 1 shows why the volumes of pieces add, and that two different ways of cutting give the same total. Day 2 finds edges the drawing does not label, and handles solids of three boxes, such as a medal podium. Day 3 takes a part away, for a planter box with walls and a hollow step stool, and uses both moves in word problems.

The procedure

Day 1: volume adds up

Here is Seojun’s step stool. It is not a box, because it has a step. But it is made of boxes.

Figure 1Seojun’s step stool is not one box, but it is made of boxes.

Imagine sawing it straight down where the step begins. The cut leaves two boxes:

a tall box A, 6 in long, 9 in deep and 12 in tall;
a low box B, 8 in long, 9 in deep and 6 in tall.

Why can we add the two volumes? Think of filling the stool with inch cubes. Every cube sits in A or in B. No cube sits in both, because the pieces do not overlap. No cube is left out, because the two pieces together make the whole stool. So

\text{cubes in the stool} = \text{cubes in A} + \text{cubes in B}

That is the key fact of this lesson. Volume is additive: when a solid is made of pieces that do not overlap, its volume is the sum of the volumes of the pieces. Writing V_A for the volume of piece A and V_B for the volume of piece B,

V = V_A + V_B

There is more than one way to cut. Seojun could also cut the stool flat, at the height of the step.

Figure 2Two ways to cut the same stool into two boxes. Both ways give 1{,}080 cubic inches.

The flat cut makes a long bottom box and a smaller top box. These are different boxes from the first cut, with different volumes. But they are filled by the very same inch cubes, just grouped another way. So the total must come out the same. That gives a free check: work the volume one way, then the other way, and the two totals should agree.

A good cut runs along an edge of a step, straight through the solid, so that every piece is a box. Any such cut works. Pick the one whose edges you can read most easily from the drawing.

Day 2: edges the drawing does not show

A drawing of a solid made of boxes rarely labels every edge. It does not have to, because edges that run the same way fit together.

Look at Seojun’s stool again. Along the bottom, the tall part (6 in) and the low step (8 in) together make the whole length, 6 + 8 = 14 in. Up the side, the low step (6 in) and the part of the tall box above it together make the whole height, 12 in, so that upper part is 12 - 6 = 6 in tall. In general:

lengths that run in the same direction and sit end to end add to the whole;
so a missing part is the whole minus the parts you know.

A third edge, the depth, runs front to back. If every part of the stool has the same depth, one label is enough for all of them.

Some solids need three boxes. A medal podium has three steps side by side: first place in the middle and tallest, second place on one side, third place on the other. Each step is its own box, so the podium splits into three boxes, and its volume is the sum of three volumes. Additivity works for any number of pieces, for the same reason: every cube is in exactly one piece.

Figure 3A three-step medal podium. The height of each of the two tall steps is written on its front.

The drawing gives the whole length, 14 ft, and the two side steps, 5 ft each. The middle step is what is left:

14 - 5 - 5 = 4 \text{ ft}

Example 4 finishes this podium.

Day 3: taking a part away

Some solids are easier to see as a whole box with a piece missing. Wapi’s planter box is the plainest case.

Figure 4Wapi’s planter box. Its four walls and its floor are boards 1 inch thick.

The outside of the planter is one big box. That big box is made of two parts that do not overlap: the wood, and the empty space inside where the soil goes. So additivity says

\text{outside box} = \text{wood} + \text{inside space}

Read backward, that is a subtraction:

\text{wood} = \text{outside box} - \text{inside space}

Subtracting is not a new rule. It is additivity used the other way around.

The planter shows one more thing. Two questions about it have two different answers.

How much soil does it hold? That is the space inside, so it uses the inside edges.
How much wood is in it? That is the outside box minus the inside space.

The inside edges come from the outside ones. Each long side loses a wall at both ends, so the inside length is the outside length minus two wall thicknesses. The same is true of the width. The height loses only the floor, because the top is open.

A hollow step stool works the same way. It is a whole box with a box-shaped tunnel taken out from front to back, so the stool stands on two legs.

Figure 5A hollow step stool. It is a whole box with a box-shaped tunnel taken out.

Its volume of wood can be found by subtracting the tunnel from the whole box, or by adding the top and the two legs. Example 7 does both and gets the same answer, as it must.

In a word problem, decide first what the question is asking for: the space a thing takes up (its outside), the space it holds (its inside), or the material it is made of (the difference). Then split or subtract.

Worked examples

Examples 1 and 2 go with Day 1, Examples 3, 4 and 5 with Day 2, and Examples 6 and 7 with Day 3.

Example 1

Seojun’s step stool, cut two ways

Seojun’s step stool, drawn above, is 14 in long and 9 in deep. The tall part is 6 in long and 12 in tall. The low step is 8 in long and 6 in tall. Find the volume of the stool. Then check it by cutting the other way.

Solution

First way: cut straight down where the step begins. The tall box is 6 by 9 by 12:

6 \times 9 = 54, \qquad 54 \times 12 = 648

The low box is 8 by 9 by 6:

8 \times 9 = 72, \qquad 72 \times 6 = 432

The two boxes do not overlap and together make the whole stool, so their volumes add:

648 + 432 = 1{,}080

Second way: cut flat at the height of the step, 6 in up. The bottom box runs the whole length. It is 14 by 9 by 6:

14 \times 9 = 126, \qquad 126 \times 6 = 756

Above the cut is only the top of the tall part. It is 6 in long and 9 in deep, and it is 12 - 6 = 6 in tall:

6 \times 9 \times 6 = 324

Adding, 756 + 324 = 1{,}080.

The stool’s volume is 1{,}080\ \text{in}^3. The two ways agree because both count the same inch cubes, grouped differently.

Example 2

A medal stand with a box on top

Next to the podium is a stand for the medals. It is a box 15 in long, 12 in deep and 4 in tall, with a box 8 in long, 6 in deep and 7 in tall set on top, in the middle. How tall is the stand, and what is its volume?

Solution

The top box sits on the bottom box, so the heights stack: 4 + 7 = 11 in tall.

The two boxes do not overlap, so the volume is the sum of their volumes. The bottom box:

15 \times 12 \times 4 = 180 \times 4 = 720

The top box:

8 \times 6 \times 7 = 48 \times 7 = 336

Adding, 720 + 336 = 1{,}056. The stand is 11 in tall and its volume is 1{,}056\ \text{in}^3.

The top box does not line up with any edge of the bottom box. That does not matter. Additivity asks only that the pieces do not overlap and together make the whole solid.

Example 3

Pooja’s step stool, with an edge to find

Pooja’s step stool is drawn below. Find the length of the low step, then the volume of the stool.

Figure 6Pooja’s step stool. The length of the low step is not labeled, but the drawing still gives it.

Solution

Along the bottom, the tall part and the low step sit end to end and make the whole length, 18 in. The tall part is 7 in, so the low step is

18 - 7 = 11 \text{ in long}

Both parts have the same depth, 10 in. Cut straight down where the step begins. The tall box is 7 by 10 by 15, and the low box is 11 by 10 by 8:

\begin{aligned} 7 \times 10 \times 15 &= 70 \times 15 = 1{,}050 \\ 11 \times 10 \times 8 &= 110 \times 8 = 880 \end{aligned}

Adding, 1{,}050 + 880 = 1{,}930. The volume is 1{,}930\ \text{in}^3.

Check. Cut flat at the height of the step instead. The bottom box is 18 by 10 by 8, which is 1{,}440. The piece above the cut is 7 in long, 10 in deep and 15 - 8 = 7 in tall, which is 490. And 1{,}440 + 490 = 1{,}930. The same.

Example 4

The medal podium

Finish the podium from Day 2. It is 4 ft deep and 14 ft long overall. The second-place step is 5 ft long and 2 ft tall. The first-place step is 3 ft tall. The third-place step is 5 ft long and 1 ft tall. What is the volume of the podium?

Solution

The first-place step is the length that is left: 14 - 5 - 5 = 4 ft. Each step is a box 4 ft deep, and the three boxes do not overlap, so add their volumes.

\begin{aligned} \text{second place:}\ 5 \times 4 \times 2 &= 40 \\ \text{first place:}\ 4 \times 4 \times 3 &= 48 \\ \text{third place:}\ 5 \times 4 \times 1 &= 20 \end{aligned}

The total is 40 + 48 + 20 = 108. The podium’s volume is 108\ \text{ft}^3.

Check. Cut the podium into flat layers 1 ft thick instead. The bottom layer runs the whole length: 14 \times 4 \times 1 = 56. The second layer, from 1 ft to 2 ft up, covers the second-place and first-place steps, 5 + 4 = 9 ft of length: 9 \times 4 \times 1 = 36. The top layer is only the first-place step: 4 \times 4 \times 1 = 16. And 56 + 36 + 16 = 108.

Example 5

A model podium, worked backward

Pooja builds a solid wooden model of a podium for a table. Its three steps sit side by side, and each is 7 in long and 5 in deep. The first-place step is 12 in tall and the second-place step is 9 in tall. The whole model is 910 cubic inches of wood. How tall is the third-place step?

Solution

Every step has the same base, 7 \times 5 = 35 square inches. The two known steps are

35 \times 12 = 420 \qquad \text{and} \qquad 35 \times 9 = 315

Together they make 420 + 315 = 735. By additivity, the whole model is these two steps plus the third, so the third step’s volume is what is left:

910 - 735 = 175 \text{ cubic inches}

The third step is a box with a base of 35 square inches. Its height is the number of layers of 35 cubes that make 175:

175 \div 35 = 5

The third-place step is 5 in tall.

Check. 420 + 315 + 35 \times 5 = 420 + 315 + 175 = 910.

There is a shorter road. Because all three steps share the base 35, the volume is 35 times the sum of the heights, by the distributive property. So the heights add to 910 \div 35 = 26, and 26 - 12 - 9 = 5.

Example 6

Wapi’s planter box

Wapi’s planter box, drawn above, is 26 in long, 10 in wide and 9 in tall on the outside. It is open at the top, and its walls and floor are boards 1 in thick.

(a) How much soil does it take to fill the planter to the top?

(b) How much wood is in the planter?

Solution

(a) The soil fills the space inside, so use the inside edges.

Length: a wall at each end, so 26 - 1 - 1 = 24 in.
Width: a wall at the front and at the back, so 10 - 1 - 1 = 8 in.
Height: only the floor, since the top is open, so 9 - 1 = 8 in.

24 \times 8 \times 8 = 192 \times 8 = 1{,}536

It takes 1{,}536\ \text{in}^3 of soil.

(b) The outside box is the wood plus the space inside, and those two do not overlap. The outside box is

26 \times 10 \times 9 = 260 \times 9 = 2{,}340

So the wood is 2{,}340 - 1{,}536 = 804. There are 804\ \text{in}^3 of wood.

The planter takes up 2{,}340\ \text{in}^3 of space on the porch but holds only 1{,}536, because the walls and floor use up the rest.

Example 7

The hollow step stool, two ways

The hollow step stool, drawn above, is a box 20 in long, 12 in deep and 10 in tall. A tunnel 14 in wide and 7 in tall is cut through it from front to back, in the middle. Find the volume of wood by subtracting. Then check by adding.

Solution

Subtracting. The whole box is the wood plus the tunnel. The whole box is

20 \times 12 \times 10 = 2{,}400

The tunnel is a box 14 by 12 by 7, since it runs the full depth:

14 \times 12 \times 7 = 168 \times 7 = 1{,}176

So the wood is 2{,}400 - 1{,}176 = 1{,}224.

Adding. Split the wood into a top and two legs. The tunnel is 7 in tall, so the top is 10 - 7 = 3 in thick, and it runs the whole length:

20 \times 12 \times 3 = 720

The tunnel takes 14 of the 20 inches of length, and it is in the middle, so each leg is (20 - 14) \div 2 = 3 in wide. Each leg is 3 by 12 by 7:

3 \times 12 \times 7 = 252, \qquad 2 \times 252 = 504

Adding, 720 + 504 = 1{,}224.

Both ways give 1{,}224\ \text{in}^3 of wood.

Boundaries

The pieces must not overlap. If two pieces share some space, that space is counted twice. For Seojun’s stool, adding the whole bottom box (756) and the whole tall part (648) gives 1{,}404, which is too much. The part where they overlap, 6 by 9 by 6, is 324, and 1{,}404 - 324 = 1{,}080, the true volume.
The pieces must make the whole solid. A piece left out is volume left out.
What is taken away must lie inside the big box. Subtracting a tunnel or a hollow works only because the tunnel is part of the whole box to begin with.
Inside for what it holds, outside for what it takes up. A container with thick walls holds less than the space it fills. The soil in a planter comes from the inside edges.
Boxes only. Every piece here is a box with square corners. A solid with a curved or slanted part, such as a silo or a ramp, needs other pieces, and Composite solids and density handles those.
One unit for every edge. Inches and feet must be made the same before multiplying.
Whole-number edges in this lesson. Splitting and adding works the same way when an edge is a fraction; Volume of a box with fraction edge lengths shows why the box rule still holds then.

Where this goes next

Splitting and subtracting are the two moves behind every volume of a solid made of pieces. Composite solids and density uses them on solids with curved and slanted parts, such as a silo made of a cylinder and a cone, where each piece has its own rule. Flat figures use the same two moves in Area of composite figures and shaded regions. Measuring the outside faces of a stool or a planter, instead of the space inside, leads to Surface area, and how it compares with volume, and boxes with other bases lead to Volume of prisms and cylinders.

Exercises

Each set goes with one day of the lesson. The last set asks you to explain.

Day 1: adding the pieces

A solid is made of two boxes that do not overlap. Box A is 5 cm by 4 cm by 3 cm, and box B is 7 cm by 4 cm by 2 cm. What is the volume of the solid?

Wapi’s step stool is a bottom box 12 in long, 10 in deep and 5 in tall. On top of it, against the back, sits a box 12 in long, 5 in deep and 5 in tall. Find the volume of the stool.

Pooja lays centimeter cubes in a layer 4 cubes long and 3 cubes wide. Along the back edge she adds a row of 4 cubes on top, and on that row one more row of 4. How many cubes does she use? What is the volume of her solid?

A step stool is 10 in long and 5 in deep. Its tall part is 4 in long and 9 in tall. Its low step is 6 in long and 3 in tall. Find its volume twice: once cutting straight down where the step begins, and once cutting flat at the height of the step.

A two-step platform beside the podium is a box 8 ft long, 3 ft deep and 2 ft tall, with a box 4 ft long, 3 ft deep and 2 ft tall set on top at one end. What is its volume?

A trophy stand is a box 10 in long, 6 in deep and 2 in tall, with a cube 3 in on each edge set on top. What is its volume?

A solid of two boxes has a volume of 520 cubic centimeters. One box is 10 cm by 4 cm by 5 cm. The other is 8 cm long and 5 cm wide. How tall is the other box?

Day 2: missing edges and three boxes

A step stool is 16 in long overall and 8 in deep. Its tall part is 6 in long and 10 in tall, and its low step is 4 in tall. How long is the low step? What is the volume?

A step stool is 12 in long and 9 in deep. Its tall part is 5 in long and 14 in tall, and its low step is 5 in tall. Cut it flat at the height of the low step. How tall is the top piece? What is the volume of the stool?

A podium has three steps side by side, each 5 ft long and 3 ft deep. The steps are 2 ft, 3 ft and 1 ft tall. What is the volume of the podium?

A podium is 16 ft long overall and 4 ft deep. The second-place and third-place steps are each 5 ft long, with the first-place step between them. First place is 4 ft tall, second place 3 ft and third place 2 ft. How long is the first-place step? What is the volume?

A stool has three steps in a row, like stairs. Each step is 6 in long and 8 in deep, and the steps are 6 in, 12 in and 18 in tall. Find the volume. Then find it again by cutting the stool into flat layers 6 in thick.

A step stool made of two boxes has a volume of 1{,}240 cubic inches. The bottom box is 14 in by 10 in by 6 in. The top box has a base 8 in by 10 in. How tall is the top box?

The three steps of a podium are each 5 ft long and 2 ft deep. The first-place step is 4 ft tall and the second-place step is 3 ft tall. The whole podium is 80 cubic feet. How tall is the third-place step?

Day 3: taking a part away, and word problems

A block of wood is 10 cm long, 8 cm deep and 7 cm tall. A box-shaped hole 4 cm wide and 3 cm tall is cut all the way through it from front to back. What volume of wood is left?

Wapi builds a second planter box, open at the top. On the outside it is 22 in long, 9 in wide and 8 in tall, and its walls and floor are 1 in thick. (a) How much soil fills it to the top? (b) How much wood is in it?

The inside of a planter box is 40 cm long, 20 cm wide and 15 cm deep. Soil comes in bags of 5{,}000 cubic centimeters. How many bags does it take to fill the box?

A step stool is a box 16 in long, 10 in deep and 12 in tall, with a tunnel 10 in wide and 8 in tall cut through it from front to back, in the middle. Find the volume of wood by subtracting, then by adding the top and the two legs.

Pooja has three planter boxes, each 30 in by 10 in by 7 in on the inside. She has 6{,}000 cubic inches of soil. Is that enough to fill all three? If not, how much more does she need?

A step stool is 12 in long, 8 in deep and 10 in tall at its tall end. Its low step leaves out a box-shaped corner 7 in long, 8 in deep and 4 in tall. Find the volume by taking the corner away from a whole box. Then check by adding two boxes.

A planter box is 28 in by 12 in by 10 in on the inside. Wapi stands a box-shaped water tank, 6 in by 12 in by 10 in, inside it at one end. How much space is left for soil?

Reasoning

Pooja cuts a stepped solid straight down and finds a volume of 400 cubic inches. Seojun cuts the same solid flat and finds 410 cubic inches. Can both be right? Explain.

Wapi finds the volume of the stool in Exercise 4 by adding the whole bottom box, 10 in by 5 in by 3 in, and the whole tall part, 4 in by 5 in by 9 in. What total does he get? Why is it too big, and by how much?

Explain why the soil a planter box holds is found from its inside edges, and why the outside volume minus the inside volume gives the wood.

Explain why two different ways of cutting the same solid into boxes must give the same total volume.

Answers

116\ \text{cm}^3. Box A is 5 \times 4 \times 3 = 60 and box B is 7 \times 4 \times 2 = 56. They do not overlap, so 60 + 56 = 116.

900\ \text{in}^3. The bottom box is 12 \times 10 \times 5 = 600 and the top box is 12 \times 5 \times 5 = 300, and 600 + 300 = 900.

20 cubes, so 20\ \text{cm}^3. The layer has 4 \times 3 = 12 cubes, and the two rows add 4 + 4 = 8, so 12 + 8 = 20.

270\ \text{in}^3 both ways. Straight down: 4 \times 5 \times 9 = 180 and 6 \times 5 \times 3 = 90, total 270. Flat: the bottom box is 10 \times 5 \times 3 = 150 and the top piece is 4 by 5 by 9 - 3 = 6, which is 120, total 270.

72\ \text{ft}^3. 8 \times 3 \times 2 = 48 and 4 \times 3 \times 2 = 24, and 48 + 24 = 72.

147\ \text{in}^3. The box is 10 \times 6 \times 2 = 120 and the cube is 3 \times 3 \times 3 = 27, and 120 + 27 = 147.

8 cm tall. The first box is 10 \times 4 \times 5 = 200, so the other is 520 - 200 = 320. Its base is 8 \times 5 = 40, and 320 \div 40 = 8.

The low step is 10 in long, and the volume is 800\ \text{in}^3. 16 - 6 = 10. The tall box is 6 \times 8 \times 10 = 480 and the low box is 10 \times 8 \times 4 = 320, and 480 + 320 = 800.

The top piece is 9 in tall, and the volume is 945\ \text{in}^3. 14 - 5 = 9. The bottom box is 12 \times 9 \times 5 = 540 and the top piece is 5 \times 9 \times 9 = 405, and 540 + 405 = 945.

90\ \text{ft}^3. Each base is 5 \times 3 = 15, so the steps are 30, 45 and 15, and 30 + 45 + 15 = 90.

The first-place step is 6 ft long, and the volume is 196\ \text{ft}^3. 16 - 5 - 5 = 6. The steps are 6 \times 4 \times 4 = 96, 5 \times 4 \times 3 = 60 and 5 \times 4 \times 2 = 40, and 96 + 60 + 40 = 196.

1{,}728\ \text{in}^3 both ways. Each step has a base 6 \times 8 = 48, so the steps are 288, 576 and 864, which add to 1{,}728. In layers: the bottom layer is 18 \times 8 \times 6 = 864, the middle 12 \times 8 \times 6 = 576 and the top 6 \times 8 \times 6 = 288, the same total.

5 in tall. The bottom box is 14 \times 10 \times 6 = 840, so the top box is 1{,}240 - 840 = 400. Its base is 8 \times 10 = 80, and 400 \div 80 = 5.

1 ft tall. Each base is 5 \times 2 = 10. The two known steps are 40 + 30 = 70, so the third is 80 - 70 = 10, and 10 \div 10 = 1.

464\ \text{cm}^3. The block is 10 \times 8 \times 7 = 560. The hole runs the full depth, so it is 4 \times 8 \times 3 = 96, and 560 - 96 = 464.

(a) 980\ \text{in}^3 of soil. The inside is 22 - 2 = 20 long, 9 - 2 = 7 wide and 8 - 1 = 7 tall, and 20 \times 7 \times 7 = 980. (b) 604\ \text{in}^3 of wood. The outside is 22 \times 9 \times 8 = 1{,}584, and 1{,}584 - 980 = 604.

3 bags. The box holds 40 \times 20 \times 15 = 12{,}000 cubic centimeters. Two bags make 10{,}000, which is not enough, so a third bag is needed.

1{,}120\ \text{in}^3 both ways. Subtracting: 16 \times 10 \times 12 = 1{,}920 minus the tunnel 10 \times 10 \times 8 = 800. Adding: the top is 12 - 8 = 4 in thick, so 16 \times 10 \times 4 = 640; each leg is (16 - 10) \div 2 = 3 in wide, so 3 \times 10 \times 8 = 240, and 640 + 240 + 240 = 1{,}120.

No. She needs 300 more cubic inches. Each box holds 30 \times 10 \times 7 = 2{,}100, so three hold 6{,}300, and 6{,}300 - 6{,}000 = 300.

736\ \text{in}^3. The whole box is 12 \times 8 \times 10 = 960 and the corner is 7 \times 8 \times 4 = 224, and 960 - 224 = 736. Adding: the tall part is 12 - 7 = 5 in long, 5 \times 8 \times 10 = 400, and the low step is 10 - 4 = 6 in tall, 7 \times 8 \times 6 = 336, and 400 + 336 = 736.

2{,}640\ \text{in}^3. The inside is 28 \times 12 \times 10 = 3{,}360 and the tank takes 6 \times 12 \times 10 = 720, and 3{,}360 - 720 = 2{,}640.

No. Both cuts break the same solid into pieces that do not overlap and that fill it, so both totals count the same unit cubes. The two answers must be equal, so at least one of them has a mistake, perhaps a wrong edge or a slip in the arithmetic.

330\ \text{in}^3, which is 60\ \text{in}^3 too big. 10 \times 5 \times 3 = 150 and 4 \times 5 \times 9 = 180 make 330, but the true volume is 270. The two boxes overlap in the bottom of the tall part, a box 4 by 5 by 3, which is 60, and that part was counted twice: 330 - 60 = 270.

The soil fills only the empty space inside, and that space is a box whose edges are the inside edges, so its volume is what the planter holds. The outside box is made of two parts that do not overlap, the wood and the inside space, so its volume is the sum of the two. Taking the inside space away from that sum leaves the wood.

Fill the solid with unit cubes. A cut that makes pieces with no overlap puts each cube in exactly one piece, so adding the pieces’ volumes counts every cube once. A different cut groups the same cubes differently, but it still counts each one once. Both totals equal the number of cubes in the solid, so they are equal.