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Grade 5 · 5.MD.C.3 · 5.MD.C.3A

Volume with unit cubes

A box is packed full of small cubes, but you can only see the outside. How can you know how many cubes are inside without taking them all out?

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Liquid volume: liters and millilitersGrade 3Area of rectangles and squaresGrade 3
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Converting square and cubic unitsMath 7Volume of a boxGrade 5Volume of prisms and cylindersMath 7
Where you are 9 in Volume and capacity
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Volume and capacity9
▴ 2 aboveCustomary capacity unitsVolume as filling
Volume with unit cubes
Volume of a boxSurface area, and how it compares with volume▾ Show all 9 (2 more)

Why this exists

Lomasi, Nikhil and Yuna have a tub of wooden cubes, each one inch along every edge, and a stack of small shipping boxes. They want to know how many cubes each box holds.

Counting one cube at a time is slow. Worse, once a box is full you can only see the cubes on the outside. The ones in the middle are hidden. In Volume as filling the volume of a solid is the number of unit cubes that fill it. For a box, the cubes are not scattered. They sit in neat rows, and the rows sit in flat layers. That order turns counting into multiplying, and it lets you know how many cubes are inside without seeing them.

The lesson runs over three days. Day 1 packs a box with whole layers of cubes. Day 2 names the units volume is measured in, and shows why they are cubed. Day 3 links the cubic centimeter to the milliliter, and shows what happens when the cube changes size.

Definition

A unit cube is a cube whose edges are each 1 unit long. Its volume is 1 cubic unit.

The volume of a box is the number of unit cubes that pack it exactly, with no gaps and no overlaps.

A layer is one flat level of cubes that covers the whole bottom of the box and is one cube thick.

The unit cube can be any size, and its name comes from its edge:

Unit cubeEdgeWrittenRead
cubic centimeter1 cm1\ \text{cm}^3one cubic centimeter
cubic inch1 in1\ \text{in}^3one cubic inch
cubic foot1 ft1\ \text{ft}^3one cubic foot
cubic meter1 m1\ \text{m}^3one cubic meter

The small raised 3 says the unit uses three lengths: length, width and height. You may also hear \text{cm}^3 read as “centimeters cubed”.

A milliliter (\text{mL}) is a unit for how much liquid a container holds, and 1{,}000 milliliters make 1 liter (see Liquid volume: liters and milliliters). Day 3 shows that

1\ \text{cm}^3 = 1\ \text{mL}.

Intuition

Day 1: a box holds whole layers

Take a box whose bottom is 4 inches long and 3 inches wide, and which is 5 inches tall. Fill it with inch cubes.

Start at the bottom. Each cube on the floor of the box sits on one square inch of the floor. The floor is a rectangle 4 by 3, so it holds 4 \times 3 = 12 unit squares, three rows of four. That means the bottom layer holds 12 cubes, one on each square.

4
3
4
5
3
\text{one layer}
Figure 1One layer covers the 4 \times 3 base with 12 cubes. Five layers fill the box, 12 \times 5 = 60 cubes.

Now stack a second layer on top. It sits on the first layer exactly as the first sat on the floor, so it also holds 12 cubes. So does the third, the fourth and the fifth. The box is 5 inches tall, and each layer is 1 inch thick, so exactly 5 layers fit. In all:

12 \times 5 = 60 \text{ cubes}.

The box has a volume of 60 cubic inches. You never had to count the hidden cubes in the middle. You knew they were there, because each layer is full and each layer is the same.

This is the whole idea of Day 1:

\begin{aligned}\text{cubes in the box} &= \text{cubes in one layer} \\ &\quad \times \text{number of layers}.\end{aligned}

And the cubes in one layer are the squares in the base. Notice what the three numbers are. The 4 and the 3 are the edges of the base, and the 5 is the number of layers, which is the height. So the 60 cubes came from 4 \times 3 \times 5.

Day 2: cubic units, and why they are cubed

Three different kinds of size come up when you measure a box.

  • How long one edge is: a length. Its unit is a short line, such as 1 centimeter.
  • How much flat surface the bottom has: an area. Its unit is a square, such as 1 square centimeter, a square 1 cm on each side.
  • How much space the box takes up: a volume. Its unit is a cube, such as 1 cubic centimeter, a cube 1 cm on each edge.
1\ \text{cm}
1\ \text{cm}^2
1\ \text{cm}^3
Figure 2A length uses one direction, an area two, a volume three.

A length goes in one direction. A square has two lengths, its length and its width, so its unit is written with a 2: \text{cm}^2. A cube has three lengths, its length, width and height, so its unit is written with a 3: \text{cm}^3. That small 3 is why we say volume units are cubed.

The different units are different sizes. A cubic centimeter is small, small enough to sit on the tip of a finger. A cubic inch is bigger: its edge is about 2\tfrac{1}{2} cm, and it is larger than an ordinary game die. A cubic foot is a box one foot each way, big enough to hold a basketball. A volume is always a count of one kind of cube, so a volume must always say which cube: “60” on its own says nothing, but 60\ \text{in}^3 is a real amount of space.

Day 3: a cubic centimeter is a milliliter

The metric system ties its volume units and its liquid units together on purpose (see The metric system and SI units). A liter is defined as the space inside a cube that is 10 centimeters on every edge.

10\ \text{cm}
10\ \text{cm}
10\ \text{cm}
Figure 3A 10 cm cube holds 10 \times 10 \times 10 = 1{,}000 centimeter cubes. It holds exactly 1 liter, so the shaded cube holds 1 milliliter.

Count that cube’s centimeter cubes with layers. The bottom is 10 by 10, so one layer holds 10 \times 10 = 100 cubes. The cube is 10 cm tall, so there are 10 layers:

100 \times 10 = 1{,}000 \text{ cubic centimeters}.

So 1 liter is 1{,}000\ \text{cm}^3. But 1 liter is also 1{,}000 milliliters. The same liter is cut into 1{,}000 equal parts both ways, so each part is the same:

1\ \text{cm}^3 = 1\ \text{mL}.

This means a box measured in centimeters tells you right away how much water it holds. A box that holds 60\ \text{cm}^3 holds 60\ \text{mL}.

Changing the cube changes the count. Suppose you pack with cubes 2 cm on each edge instead of 1 cm. Each big cube is 2 small cubes long, 2 wide and 2 tall, so it is 2 \times 2 \times 2 = 8 centimeter cubes.

1\ \text{cm}
2\ \text{cm}
2\ \text{cm}
2\ \text{cm}
Figure 4A 2 cm cube is 2 \times 2 \times 2 = 8 centimeter cubes, not 2.

A box that holds 8 centimeter cubes holds just 1 of the bigger cubes. The space has not changed. Only the unit has, so the count changes. A bigger unit cube gives a smaller count, and a cube with twice the edge is worth 8 times as much, not 2 times.

Results

Layers. A box packed with unit cubes holds

\begin{aligned}&\text{cubes in one layer} \\ &\quad \times \text{number of layers}\end{aligned}

unit cubes, and the cubes in one layer are the unit squares in the base.

Cubic units. Volume is measured in cubic units (\text{cm}^3, \text{in}^3, \text{ft}^3, \text{m}^3), each a cube one unit long in all three directions. Area is measured in square units.

Centimeters and milliliters. 1\ \text{cm}^3 = 1\ \text{mL}, and 1{,}000\ \text{cm}^3 = 1 liter.

Bigger cubes from smaller ones. A cube whose edge is n small units is made of n \times n \times n small unit cubes.

Why it is true

Layers. Every cube in a packed box is at one level, so it belongs to exactly one layer. Inside its layer it sits above exactly one square of the base. So when you count the cubes layer by layer, each cube is counted once: none missed, none counted twice. Each layer covers the base, so each layer has as many cubes as the base has squares. The walls of a box go straight up, so every layer is the same, and the number of layers is the height in cube edges. Adding the same layer count again and again is multiplying, so the total is cubes per layer times layers.

Why the units are cubed. A cube has three edges meeting at each corner, going in three directions. A cube of edge n small units has n rows of n cubes in each layer, and n layers, so it holds n \times n \times n small cubes. Each of the three directions brings one factor of n. A square of side n uses only two directions, so it holds n \times n small squares.

Centimeters and milliliters. A liter is the space inside a 10 cm cube. That cube holds 1{,}000 centimeter cubes, by layers. A milliliter is one thousandth of a liter. One thousandth of the 10 cm cube is one of its 1{,}000 centimeter cubes, so 1\ \text{mL} is exactly 1\ \text{cm}^3. Nothing was measured here: it is true because of how the liter was set up.

Worked examples

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

Example 1

A small shipping box

Lomasi fills a small shipping box with inch cubes. The bottom layer has 2 rows of 5 cubes, and the box is 3 cubes tall. How many cubes fill the box, and what is its volume?

Solution

The bottom layer has 2 rows of 5, so it holds 5 \times 2 = 10 cubes. These are the 10 square inches of the floor, one cube on each.

The box is 3 cubes tall, so 3 layers fit, each with 10 cubes:

10 \times 3 = 30 \text{ cubes}.

Each cube is 1 cubic inch, so the volume is 30\ \text{in}^3.

Check. Look down on the box instead. Each of the 10 squares of the floor has a column of 3 cubes standing on it, and 10 columns of 3 is again 30.

Example 2

Standing the box on its end

Nikhil packs a box with cubes, 6 cubes long, 3 cubes wide and 2 cubes tall. Then he tapes it shut and stands it on its small 3 by 2 end. Does the number of cubes inside change? Count it both ways.

Solution

Lying flat, the base is 6 by 3, so one layer holds 6 \times 3 = 18 cubes. The box is 2 cubes tall:

18 \times 2 = 36 \text{ cubes}.

Standing on its end, the base is 3 by 2, so one layer holds 3 \times 2 = 6 cubes. Now the long edge points up, so there are 6 layers:

6 \times 6 = 36 \text{ cubes}.

The count is the same, as it must be: nothing went in or out. Turning the box only changes which face we call the base. In numbers, (6 \times 3) \times 2 = 6 \times (3 \times 2) is the associative property of multiplication (grouping the factors differently), and choosing which two edges to multiply first is the commutative property (reordering them). Standing it on its 6 by 2 side would give 12 \times 3 = 36 too.

Example 3

Halfway through packing

Yuna is packing a box whose base holds 7 cubes by 4 cubes. The box is 3 layers tall. So far she has filled the bottom layer and put 2 rows of 7 cubes into the second layer. How many more cubes does she need to fill the box?

Solution

One full layer has 4 rows of 7, so 7 \times 4 = 28 cubes. A full box has 3 layers:

28 \times 3 = 84 \text{ cubes}.

She has placed one full layer and 2 rows of the next: 28 + 2 \times 7 = 28 + 14 = 42 cubes. She still needs

84 - 42 = 42 \text{ cubes}.

Check. Count what is missing directly. The second layer is missing 4 - 2 = 2 rows, which is 14 cubes, and the whole third layer, 28 cubes, is empty: 14 + 28 = 42.

Example 4

Lining the shoebox or filling it

Yuna’s shoebox measures 12 inches long, 7 inches wide and 5 inches tall on the inside.

(a) How much paper does it take to cover the inside bottom exactly?

(b) How many inch cubes would fill the box?

Solution

(a) The paper covers a flat surface, so this is an area, counted in square inches. The bottom is a 12 by 7 rectangle, so it is 12 \times 7 = 84 unit squares: 84\ \text{in}^2 of paper.

(b) The cubes fill space, so this is a volume, counted in cubic inches. One layer of cubes covers the bottom, one cube on each of the 84 squares, so a layer has 84 cubes. The box is 5 inches tall, so 5 layers fit:

84 \times 5 = 420 \text{ cubes}.

The volume is 420\ \text{in}^3.

The two answers measure different things. It makes no sense to say 420 is bigger than 84: one counts squares of paper, the other counts cubes of space, the way 5 apples is not more than 3 miles.

Example 5

How many cubic inches in a cubic foot?

A cubic foot is a cube 1 foot on each edge. How many cubic inches is that?

Solution

One foot is 12 inches, so the cube is 12 inches long, 12 inches wide and 12 inches tall. Pack it with inch cubes. The bottom is 12 by 12, so one layer holds 12 \times 12 = 144 cubes. There are 12 layers:

\begin{aligned}144 \times 12 &= 1{,}440 + 288 \\ &= 1{,}728 \text{ cubes}.\end{aligned}

So 1\ \text{ft}^3 = 1{,}728\ \text{in}^3.

Compare a square foot: it is only the bottom layer’s worth, 12 \times 12 = 144 square inches. A foot is 12 inches, a square foot is 12 \times 12 square inches, and a cubic foot is 12 \times 12 \times 12 cubic inches. Each new direction brings another factor of 12. That is why the units are cubed.

Example 6

How much does the box hold?

Nikhil has a plastic box that measures 25 cm long, 20 cm wide and 12 cm tall on the inside. How many milliliters of water does it hold? How many liters is that?

Solution

Think of packing it with centimeter cubes. The base is 25 by 20, so one layer holds 25 \times 20 = 500 cubes. The box is 12 cm tall, so 12 layers fit:

500 \times 12 = 6{,}000\ \text{cm}^3.

Each cubic centimeter holds 1 milliliter, so the box holds 6{,}000\ \text{mL}. A liter is 1{,}000 mL, and 6{,}000 \div 1{,}000 = 6, so that is 6 liters.

Example 7

Packing with bigger cubes

Lomasi has a small shipping box 10 cm long, 8 cm wide and 4 cm tall inside. She packs it with cubes that are 2 cm on each edge. How many of these cubes fit? What is the volume of the box in cubic centimeters, and how many milliliters does it hold?

Solution

Along the 10 cm length, 10 \div 2 = 5 cubes fit. Along the 8 cm width, 8 \div 2 = 4. Up the 4 cm height, 4 \div 2 = 2. So one layer holds 5 \times 4 = 20 big cubes, and there are 2 layers:

20 \times 2 = 40 \text{ big cubes}.

Each big cube is 2 \times 2 \times 2 = 8 cubic centimeters. So the volume is

40 \times 8 = 320\ \text{cm}^3,

and the box holds 320\ \text{mL}.

Check with centimeter cubes. One layer covers the 10 by 8 base, 80 cubes, and 4 layers fit: 80 \times 4 = 320. The two counts, 40 and 320, describe the same space. The edge of the cube was doubled, and the count went down 8 times, because 320 \div 40 = 8.

Boundaries

  • Whole layers need whole edges. The box fills exactly only when each edge is a whole number of cube edges. A box 5 cm long cannot be packed exactly with 2 cm cubes: two fit and a 1 cm gap is left. Then you need a smaller cube, which is what Volume of a box with fraction edge lengths does with cubes of edge \tfrac{1}{2} or \tfrac{1}{4}.
  • Layers need straight sides. Every layer is the same only when the walls go straight up from a rectangular floor, as in a box with square corners. In a bowl or a pyramid the layers shrink as you go up, so cubes per layer times layers does not apply. Counting cubes in other solids belongs to Volume as filling.
  • Inside, not outside. What a box holds is set by its inside measurements. Thick walls take up some of the space measured on the outside.
  • Space, not weight. 1\ \text{cm}^3 = 1\ \text{mL} is about space and is exact for anything that fills it, water or sand or air. It says nothing about how heavy the stuff is.
  • One kind of unit at a time. A volume counts cubes of one size. Square units and cubic units measure different things and are never added or compared, and a volume with no unit is not an answer.

Where this goes next

Counting cubes per layer times layers works, but it asks you to picture the cubes every time. The next lesson, Volume of a box, notices that cubes per layer is always the length times the width, and that the number of layers is always the height. It turns that into a rule for the volume of any box from its three edges, and uses the rule backward to find a missing edge.

Exercises

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

Day 1: layers in a box

  1. A box has a base of 7 cubes by 2 cubes and is 3 layers tall. How many cubes fill it?
  1. A box has a base of 8 cubes by 3 cubes and is 6 layers tall. How many cubes fill it?
  1. A full box holds 45 cubes, and each layer has 9 cubes. How many layers are there?
  1. Nikhil packs a box whose base is 2 cubes by 9 cubes, 5 layers tall. Count the cubes with layers. Then stand the box on its 2 by 5 face and count again.
  1. A box has a base of 6 cubes by 5 cubes and is 4 layers tall. Yuna has put in 2 full layers and 4 rows of 6 cubes in the third layer. How many more cubes does she need?
  1. A full box holds 96 cubes in 4 layers. Each row has 8 cubes. How many rows are in each layer?

Day 2: cubic units

  1. Would each answer be in units, square units or cubic units? Explain each choice. (a) The height of a shipping box. (b) The cardboard needed for its lid. (c) The space inside it.
  1. A shipping box is packed with inch cubes, 10 along its length, 6 along its width and 4 up its height. What is its volume?
  1. A garden shed has a floor 6 feet by 5 feet and is 7 feet tall inside. (a) What is the area of the floor? (b) What is the volume of the shed?
  1. A yard is 3 feet. How many cubic feet are in a cubic yard?
  1. A meter is 100 centimeters. How many cubic centimeters are in a cubic meter? How many liters does a cubic meter hold?
  1. A box is 2 feet long, 1 foot wide and 1 foot tall. What is its volume in cubic feet? In cubic inches?
  1. Box A is 3 ft by 2 ft by 2 ft. Box B is 6 ft by 2 ft by 1 ft. Which holds more?

Day 3: cubic centimeters and milliliters

  1. A box is 10 cm by 5 cm by 4 cm inside. How many milliliters does it hold?
  1. A container is 20 cm by 10 cm by 10 cm inside. How many milliliters does it hold? How many liters?
  1. A fish tank is 30 cm long, 20 cm wide and 25 cm tall inside. How many liters does it hold when full?
  1. Lomasi pours 500 mL of water into an empty box whose base is 10 cm by 10 cm. How deep is the water?
  1. A box with a base of 25 cm by 10 cm holds exactly 1 liter when full. How tall is it inside?
  1. A cube is 3 cm on each edge. How many centimeter cubes make it? How many milliliters does it hold?
  1. A box 12 cm by 8 cm by 6 cm is packed with cubes 2 cm on each edge. How many cubes fit? What is the volume in cubic centimeters?
  1. A cube is 6 cm on each edge. How many 3 cm cubes pack it? How many centimeter cubes?

Reasoning

  1. Explain why a cubic foot is not 12 cubic inches.
  1. Nikhil says that if you double every edge of a box, its volume doubles. Test it on a box 2 by 1 by 1, then explain what really happens.
  1. Explain why the bottom layer of a packed box holds exactly as many cubes as the base has unit squares.
  1. Yuna has a box 5 cm by 4 cm by 3 cm inside. How many 2 cm cubes fit in it? Explain why that count does not give the volume of the box.
Answers
  1. 42 cubes. One layer is 7 \times 2 = 14 cubes, and 14 \times 3 = 42.
  1. 144 cubes. One layer is 8 \times 3 = 24 cubes, and 24 \times 6 = 144.
  1. 5 layers. Each layer adds 9 cubes, and 45 \div 9 = 5.
  1. 90 cubes both ways. Lying flat: one layer is 2 \times 9 = 18, and 18 \times 5 = 90. Standing on the 2 by 5 face: one layer is 10, the 9 edge points up, and 10 \times 9 = 90.
  1. 36 more cubes. A layer is 6 \times 5 = 30, so a full box is 30 \times 4 = 120. She has placed 60 + 4 \times 6 = 84, and 120 - 84 = 36.
  1. 3 rows. Each layer holds 96 \div 4 = 24 cubes, and 24 \div 8 = 3 rows of 8.
  1. (a) Units: a height is a length. (b) Square units: the lid is a flat surface, an area. (c) Cubic units: the space inside is a volume.
  1. 240\ \text{in}^3. One layer is 10 \times 6 = 60 cubes, and 60 \times 4 = 240.
  1. (a) 30 square feet, since 6 \times 5 = 30. (b) 210\ \text{ft}^3: the floor holds a layer of 30 foot cubes, 7 layers fit, and 30 \times 7 = 210.
  1. 27\ \text{ft}^3. The cubic yard is 3 feet each way: one layer is 3 \times 3 = 9 foot cubes, and 9 \times 3 = 27.
  1. 1{,}000{,}000\ \text{cm}^3, which is 1{,}000 liters. One layer is 100 \times 100 = 10{,}000 centimeter cubes, 100 layers fit, and 10{,}000 \times 100 = 1{,}000{,}000. Then 1{,}000{,}000 \div 1{,}000 = 1{,}000 liters.
  1. 2\ \text{ft}^3, which is 3{,}456\ \text{in}^3. In feet: 2 \times 1 = 2 per layer, 1 layer. In inches the box is 24 by 12 by 12: 24 \times 12 = 288 per layer, and 288 \times 12 = 3{,}456. Also 2 \times 1{,}728 = 3{,}456.
  1. They hold the same, 12\ \text{ft}^3 each. Box A: 3 \times 2 = 6 per layer, 2 layers, 12. Box B: 6 \times 2 = 12 per layer, 1 layer, 12.
  1. 200 mL. One layer is 10 \times 5 = 50 centimeter cubes, and 50 \times 4 = 200\ \text{cm}^3.
  1. 2{,}000 mL, which is 2 liters. One layer is 20 \times 10 = 200, and 200 \times 10 = 2{,}000.
  1. 15 liters. One layer is 30 \times 20 = 600 centimeter cubes, 600 \times 25 = 15{,}000\ \text{cm}^3 = 15{,}000 mL, and 15{,}000 \div 1{,}000 = 15.
  1. 5 cm deep. Each centimeter of depth is a layer of 10 \times 10 = 100\ \text{cm}^3, which is 100 mL, and 500 \div 100 = 5 layers.
  1. 4 cm tall. One liter is 1{,}000\ \text{cm}^3. One layer is 25 \times 10 = 250 cubes, and 1{,}000 \div 250 = 4 layers.
  1. 27 centimeter cubes, so 27 mL. One layer is 3 \times 3 = 9, and 9 \times 3 = 27.
  1. 72 cubes, and 576\ \text{cm}^3. Along the edges 12 \div 2 = 6, 8 \div 2 = 4 and 6 \div 2 = 3, so 6 \times 4 = 24 per layer and 24 \times 3 = 72. Each is 8\ \text{cm}^3, and 72 \times 8 = 576. Check: 8 \times 12 = 96 per centimeter layer, 96 \times 6 = 576.
  1. 8 cubes of 3 cm, or 216 centimeter cubes. Two 3 cm cubes fit along each edge, so 2 \times 2 = 4 per layer and 2 layers, 8. In centimeters: 6 \times 6 = 36 per layer, 6 layers, 216. Check: each 3 cm cube is 27\ \text{cm}^3, and 8 \times 27 = 216.
  1. A cubic foot is 12 inches long, 12 wide and 12 tall. Twelve inch cubes make only one row along one edge. A layer needs 12 such rows, 144 cubes, and the cube needs 12 layers, so it is 12 \times 12 \times 12 = 1{,}728 cubic inches.
  1. The volume becomes 8 times as big, not 2 times. The 2 by 1 by 1 box holds 2 cubes. Doubled, it is 4 by 2 by 2: 4 \times 2 = 8 per layer and 2 layers, 16 cubes, and 16 \div 2 = 8. Doubling the length and the width makes each layer 2 \times 2 = 4 times as big, and doubling the height makes twice as many layers, so 4 \times 2 = 8.
  1. Each cube in the bottom layer rests on the floor and covers exactly one unit square of it, since a cube’s bottom face is a unit square. The layer covers the whole floor with no gaps and no overlaps, so every square has exactly one cube on it and every cube is on one square. The cubes and the squares match one for one, so there are the same number.
  1. 4 cubes. Along the 5 cm edge only 2 fit, with 1 cm left over; along the 4 cm edge 2 fit; up the 3 cm edge only 1 fits, with 1 cm left over. So 2 \times 2 = 4 cubes fill 4 \times 8 = 32\ \text{cm}^3, but the box is 5 \times 4 = 20 per layer times 3 layers, 60\ \text{cm}^3. The count misses the gaps, so it is not the volume. The edges are not whole numbers of 2 cm, so this cube cannot pack the box exactly.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0