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Volume of Boxes: Count Cubes in Layers, Then Multiply

Find the volume of rectangular prisms: count 1 cm cubes in one layer, multiply by the number of layers, use l × w × h and write the answer in cm³.

Grades 4–6 · 30 practice questions · Original material
Learning objectiveFind the volume of a rectangular prism by counting the cubes in one layer and multiplying by the number of layers, connect this to length × width × height, write volume in cubic units (cm³, m³), find a missing edge from the volume, and tell volume apart from area and perimeter.

The idea

Perimeter is the distance around a flat shape, and area counts the squares that cover it. Volume goes one step further: it measures how much space a 3-D object takes up, such as a box, a block or a fish tank. We measure volume by counting cubes.

A centimetre cube is 1 cm long, 1 cm wide and 1 cm tall. Its volume is 1 cubic centimetre, written 1 cm³. The small 3 means three directions: length, width and height. A cube 1 m on each edge has a volume of 1 m³, which is big enough for a child to sit inside.

Counting every cube one by one is slow, and in a drawing many cubes are hidden at the back or in the middle. Count in layers instead. The bottom layer of a box that is 4 cm long and 3 cm wide holds 4 × 3 = 12 cubes. If the box is 2 cm tall, there are 2 layers, so 12 × 2 = 24 cubes. The volume is 24 cm³.

That is why volume = length × width × height. Length × width counts the cubes in one layer, and height counts the layers. You may multiply in any order, because any face can be the bottom: 4 × 3 × 2, 2 × 4 × 3 and 3 × 2 × 4 all make 24.

Volume is not the same as area or perimeter. The bottom of that box has an area of 12 cm² and a perimeter of 14 cm, but the box holds 24 cm³. Different boxes can hold the same amount: a 6 × 2 × 2 box and an 8 × 3 × 1 box also hold 24 cubes. Volume also links to the capacity lesson: 1 cm³ holds 1 mL of water, so 1000 cm³ holds 1 L.

Worked examples

Finding the volume of a box by counting cubes in layersThree panels of 1 centimetre cubes. Panel 1: one layer 4 cubes long and 3 cubes wide holds 4 times 3, which is 12 cubes. Panel 2: two of those layers are stacked to make a box 4 cm long, 3 cm wide and 2 cm tall. 12 cubes per layer times 2 layers is 24 cubes, so the volume is 24 cubic centimetres. Panel 3: a different box, 6 cm long, 2 cm wide and 2 cm tall, also holds 6 times 2 times 2, which is 24 cubic centimetres. Volume equals length times width times height, measured in cubic centimetres.Volume: count one layer, then multiply by the layersEach small cube is 1 cm long, 1 cm wide and 1 cm tall: 1 cm³. Colours show layers. Original schematic.1 · Count one layer4 cm3 cm1 cm2 · Stack the layers4 cm3 cm2 cmlayer 2layer 13 · Same volume, new box6 cm2 cm2 cm4 × 3 = 12 cubes12 × 2 = 24 cm³6 × 2 × 2 = 24 cm³Volume = length × width × height. Length × width counts one layer; height counts the layers.Volume uses cubic units (cm³). The bottom of box 2 has an area of 12 cm², but the box holds 24 cm³.
Original schematic of 1 cm cubes, not drawn to scale. Count the cubes in one layer (length × width), then multiply by the number of layers (height). Hidden cubes at the back and in the middle still count. Boxes with different shapes can have the same volume.

Example 1 · Count in layers

A box is 4 cm long, 3 cm wide and 2 cm tall. How many 1 cm cubes fill it?

  1. Bottom layer: 4 rows of 3 cubes, or 4 × 3 = 12 cubes.
  2. The box is 2 cm tall, so there are 2 layers.
  3. 12 × 2 = 24 cubes.

24 cubes, so the volume is 24 cm³.

Example 2 · Use length × width × height

A tissue box is 12 cm long, 10 cm wide and 9 cm tall. Find its volume.

  1. Multiply length × width: 12 × 10 = 120 cubes in one layer.
  2. Multiply by the height: 120 × 9 = 1080.
  3. Write the cubic unit.

1080 cm³

Example 3 · Find a missing edge

A box has a volume of 60 cm³. Its bottom is 5 cm by 4 cm. How tall is it?

  1. One layer holds 5 × 4 = 20 cubes.
  2. How many layers make 60? 60 ÷ 20 = 3.
  3. Check: 5 × 4 × 3 = 60.

The box is 3 cm tall.

Example 4 · Same volume, different boxes

Show that a 6 × 2 × 2 box and an 8 × 3 × 1 box hold the same number of cubes as a 4 × 3 × 2 box.

  1. 4 × 3 × 2 = 24.
  2. 6 × 2 × 2 = 24.
  3. 8 × 3 × 1 = 24.

All three hold 24 cm³, even though they look different.

Example 5 · Two blocks joined

A step is made of a bottom block 5 cm × 2 cm × 1 cm with a 2 cm × 2 cm × 1 cm block on top at one end. Find the total volume.

  1. Split the solid into two boxes, the same way we split L-shapes for area.
  2. Bottom: 5 × 2 × 1 = 10 cm³. Top: 2 × 2 × 1 = 4 cm³.
  3. Add: 10 + 4 = 14 cm³.

14 cm³

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Adding the three edges instead of multiplying.

Why this fails: 4 + 3 + 2 = 9 does not count cubes. Length × width counts one layer and the height counts the layers: 4 × 3 × 2 = 24.

Counting only the cubes you can see in a drawing.

Why this fails: Cubes at the back, bottom and middle are hidden. Count one layer, then multiply by the number of layers.

Stopping after one layer.

Why this fails: Length × width gives the area of the bottom, which is only the first layer. Multiply by the height to count every layer.

Writing cm or cm² for volume.

Why this fails: Volume counts cubes, so it uses cubic units: cm³ or m³. cm² is for area and cm is for length.

Thinking a longer box must hold more.

Why this fails: One long edge does not decide volume. Multiply all three edges: a 5 × 5 × 2 box (50 cm³) holds more than a 6 × 4 × 2 box (48 cm³).

Foundation

Foundation practice

Count cubes in layers and use length × width × height.

One layer

1. A flat block is one layer of 1 cm cubes, 3 cubes long and 2 cubes wide. What is its volume?

  1. 5 cm³
  2. 6 cm³
  3. 10 cm³
  4. 6 cm
Count layers

2. A box has a bottom layer of 5 by 4 cubes. It is filled with 3 equal layers. How many cubes are in the box?

  1. 12
  2. 20
  3. 23
  4. 60
Units

3. Which unit could measure the volume of a shoebox?

  1. cm³
  2. cm²
  3. kg
  4. cm
Use the formula

4. A box is 6 cm long, 2 cm wide and 3 cm tall. What is its volume?

  1. 11 cm³
  2. 12 cm³
  3. 36 cm³
  4. 72 cm³
Layers

5. The bottom layer of a box holds 8 cubes. The box is 5 layers tall. How many cubes fill it?

  1. 13
  2. 8
  3. 40
  4. 32
Choose the measure

6. Which job needs volume?

  1. Finding how much sand fills a sandbox
  2. Covering the floor of a room with carpet
  3. Putting a fence around a sandbox
  4. Measuring how tall a door is
Order

7. Lin finds a box's volume with 2 × 5 × 3. Omar uses 5 × 3 × 2 for the same box. Who is right?

  1. Only Lin
  2. Only Omar
  3. Neither: you must add the edges
  4. Both: the volume is 30 cm³
Cube

8. A cube has edges 3 cm long. What is its volume?

  1. 9 cm³
  2. 27 cm³
  3. 12 cm³
  4. 6 cm³

Practice

Practice practice

Find volumes, missing edges and the volume of two joined blocks.

Find the volume

9. A wooden block is 7 cm long, 3 cm wide and 2 cm tall. What is its volume?

  1. 42 cm³
  2. 21 cm³
  3. 12 cm³
  4. 24 cm³
Find the volume

10. A storage box is 10 cm long, 5 cm wide and 4 cm tall. What is its volume?

  1. 19 cm³
  2. 50 cm³
  3. 200 cm³
  4. 2000 cm³
Metres

11. A garden bed is 3 m long and 2 m wide. It is filled with soil 1 m deep. How much soil does it hold?

  1. 6 m²
  2. 12 m³
  3. 5 m³
  4. 6 m³
Missing edge

12. A box has a volume of 48 cm³. Its bottom is 4 cm by 3 cm. How tall is it?

  1. 12 cm
  2. 4 cm
  3. 41 cm
  4. 6 cm
Missing edge

13. A box has a volume of 60 cm³. It is 5 cm long and 3 cm tall. How wide is it?

  1. 12 cm
  2. 4 cm
  3. 52 cm
  4. 15 cm
Compare

14. Box A is 6 cm × 4 cm × 2 cm. Box B is 5 cm × 5 cm × 2 cm. Which holds more 1 cm cubes?

  1. Box A, by 2 cubes
  2. Box A, because 6 is the longest edge
  3. They hold the same
  4. Box B, by 2 cubes
Two blocks

15. A step is built from cubes. The bottom block is 5 cm long, 2 cm wide and 1 cm tall. On one end sits a top block 2 cm long, 2 cm wide and 1 cm tall. What is the total volume?

  1. 14 cm³
  2. 20 cm³
  3. 4 cm³
  4. 10 cm³
Fill a box

16. How many 1 cm cubes fill a box 6 cm long, 4 cm wide and 3 cm tall?

  1. 13
  2. 24
  3. 72
  4. 48

Think

Think practice

Each question shows a tempting mistake. Explain why it fails.

Spot the error

17. Mia says a 4 cm × 3 cm × 2 cm box has a volume of 4 + 3 + 2 = 9 cm³. What went wrong?

  1. Nothing; volume adds the edges
  2. She should double the sum: 18 cm³
  3. She should multiply only two edges: 4 × 3 = 12 cm³
  4. She should multiply: 4 × 3 × 2 = 24 cm³
Hidden cubes

18. A 3 × 3 × 3 cube made of small cubes is drawn on paper. Ben counts the small cubes he can see and gets 19. How many small cubes are there?

  1. 19
  2. 27
  3. 9
  4. 54
Units

19. Raj writes the volume of a 4 cm × 3 cm × 2 cm box as 24 cm². What should he fix?

  1. Nothing, it is correct
  2. Change the number to 9
  3. Change cm² to cm³
  4. Change cm² to cm
Same volume?

20. Which box does NOT have a volume of 24 cm³?

  1. 5 cm × 3 cm × 2 cm
  2. 8 cm × 3 cm × 1 cm
  3. 6 cm × 2 cm × 2 cm
  4. 12 cm × 2 cm × 1 cm
Change one edge

21. A box is 3 cm × 2 cm × 4 cm. Its height is doubled to 8 cm. What happens to its volume?

  1. It stays the same
  2. It goes up by 4 cm³
  3. It doubles, from 24 cm³ to 48 cm³
  4. It becomes four times as big
Change every edge

22. A 1 cm cube is made into a bigger cube by doubling every edge to 2 cm. How many 1 cm cubes does the bigger cube hold?

  1. 8
  2. 4
  3. 6
  4. 2
Spot the error

23. A box is 5 cm long, 4 cm wide and 2 cm tall. Sam says, "The bottom has an area of 20 cm², so the volume is 20 cm³." What did Sam forget?

  1. He should use the perimeter: 18 cm³
  2. He should add 2: the volume is 22 cm³
  3. Nothing; area and volume are the same here
  4. He forgot the second layer: the volume is 40 cm³
Explain

24. Explain why length × width × height counts the cubes in a box. Use a 4 cm × 3 cm × 5 cm box in your answer.

Challenge

Challenge practice

Use volume with bigger cubes, litres and real objects.

Bigger cubes

25. How many 2 cm cubes fit exactly in a box 6 cm long, 4 cm wide and 4 cm tall?

  1. 96
  2. 12
  3. 48
  4. 24
Litres

26. An aquarium is 50 cm long, 30 cm wide and 20 cm tall. 1000 cm³ is the same as 1 L. How many litres does it hold when full?

  1. 3 L
  2. 100 L
  3. 30 L
  4. 300 L
Real box

27. A cereal box is 20 cm wide, 7 cm deep and 30 cm tall. What is its volume?

  1. 4200 cm³
  2. 57 cm³
  3. 140 cm³
  4. 600 cm³
Work backwards

28. A cube has a volume of 64 cm³. How long is each edge?

  1. 21 cm
  2. 8 cm
  3. 16 cm
  4. 4 cm
Take some away

29. A block of cubes is 5 cm long, 4 cm wide and 3 cm tall. A 2 cm by 2 cm hole is cut straight down through the whole block. What volume is left?

  1. 60 cm³
  2. 56 cm³
  3. 12 cm³
  4. 48 cm³
Design

30. Find three different boxes with whole-centimetre edges that each have a volume of 36 cm³. Write each as length × width × height.

Answer key

  1. 1. 6 cm³
    3 × 2 = 6 cubes, and each cube is 1 cm³, so 6 cm³. 5 cm³ adds 3 + 2 instead of multiplying, 10 cm³ is the perimeter number of the bottom, and 6 cm is a length unit, not a volume unit.
  2. 2. 60
    One layer is 5 × 4 = 20 cubes, and 3 layers make 20 × 3 = 60. 12 adds 5 + 4 + 3, 20 counts only the bottom layer, and 23 adds the number of layers instead of multiplying.
  3. 3. cm³
    Volume counts cubes, so it uses cubic units such as cm³. cm measures a length, cm² measures a flat area, and kg measures mass, not space.
  4. 4. 36 cm³
    6 × 2 × 3 = 36 cm³. 11 adds the three edges, 12 finds only the bottom layer (6 × 2) and 72 multiplies by 2 one time too many.
  5. 5. 40
    8 cubes per layer × 5 layers = 40 cubes. 13 adds 8 + 5, 8 is only one layer, and 32 counts only 4 layers.
  6. 6. Finding how much sand fills a sandbox
    Filling something measures the space inside a 3-D object, which is volume. A fence needs perimeter, carpet covers a flat area, and a door's height is a single length.
  7. 7. Both: the volume is 30 cm³
    You can multiply the three edges in any order: 2 × 5 × 3 = 5 × 3 × 2 = 30. The order only changes which face you call the bottom layer. Adding edges does not count cubes.
  8. 8. 27 cm³
    3 × 3 × 3 = 27 cm³. 9 cm³ is only one layer (one face), 12 adds four edges, and 6 multiplies 3 × 2.
  9. 9. 42 cm³
    7 × 3 = 21 cubes in one layer, and 2 layers make 42 cm³. 21 forgets the second layer, 12 adds the edges, and 24 is a multiplication slip.
  10. 10. 200 cm³
    10 × 5 = 50 cubes per layer, and 4 layers make 200 cm³. 19 adds the edges, 50 is one layer, and 2000 has an extra zero.
  11. 11. 6 m³
    3 × 2 × 1 = 6 m³. The answer is 6 m³, not 6 m², because the soil fills a space with depth. 5 adds 3 + 2, and 12 doubles by mistake.
  12. 12. 4 cm
    The bottom layer holds 4 × 3 = 12 cubes. 48 ÷ 12 = 4 layers, so the box is 4 cm tall. 12 cm is the layer size, not the height; 41 subtracts 4 + 3 from 48; 6 cm would give 72 cm³.
  13. 13. 4 cm
    5 × 3 = 15, and 60 ÷ 15 = 4 cm. 12 cm divides 60 by 5 only, 52 subtracts 5 + 3, and 15 is the product of the two known edges.
  14. 14. Box B, by 2 cubes
    A: 6 × 4 × 2 = 48 cm³. B: 5 × 5 × 2 = 50 cm³. B holds 2 more. Both boxes have edges that add to 12 cm, but adding edges does not compare volume, and one long edge does not decide it.
  15. 15. 14 cm³
    Split it into two boxes: 5 × 2 × 1 = 10 and 2 × 2 × 1 = 4. Add them: 10 + 4 = 14 cm³. 10 and 4 each count only one block, and 20 treats the whole step as a 5 × 2 × 2 box.
  16. 16. 72
    One layer is 6 × 4 = 24 cubes; 3 layers make 72. 13 adds the edges, 24 is one layer, and 48 is only two layers.
  17. 17. She should multiply: 4 × 3 × 2 = 24 cm³
    Volume counts cubes: 4 × 3 = 12 in each layer, times 2 layers. Adding edges does not count cubes, 12 counts only one layer, and doubling the sum is still adding.
  18. 18. 27
    3 × 3 × 3 = 27. Ben could not see the cubes hidden at the back, bottom and middle (8 of them). 9 is one layer, and 54 multiplies by 2 for no reason.
  19. 19. Change cm² to cm³
    24 is right, but volume counts cubes, so the unit is cm³. cm² is for flat area and cm is for length. 9 would come from adding the edges.
  20. 20. 5 cm × 3 cm × 2 cm
    5 × 3 × 2 = 30 cm³. The others all make 24: 6 × 2 × 2, 8 × 3 × 1 and 12 × 2 × 1. Different-looking boxes can have the same volume.
  21. 21. It doubles, from 24 cm³ to 48 cm³
    Doubling the height doubles the number of layers, so the volume doubles: 24 → 48 cm³. It cannot stay the same, adding 4 confuses cm with cm³, and four times would need two edges doubled.
  22. 22. 8
    2 × 2 × 2 = 8. Doubling one edge doubles the volume, but doubling all three multiplies it by 2 × 2 × 2. 2 doubles only one edge, 4 doubles two, and 6 is the number of faces.
  23. 23. He forgot the second layer: the volume is 40 cm³
    The bottom area tells how many cubes are in one layer (20). The box is 2 layers tall, so 20 × 2 = 40 cm³. Adding 2 does not count a layer, area and volume measure different things, and perimeter does not count cubes.
  24. 24. Length × width (4 × 3 = 12) counts the cubes in one layer on the bottom. The height (5) tells how many layers are stacked. 12 × 5 = 60, so the box holds 60 one-centimetre cubes and its volume is 60 cm³.
    A strong answer names the layer (length × width), the number of layers (height) and gives the unit cm³.
  25. 25. 12
    Along each edge, count 2 cm cubes: 6 ÷ 2 = 3, 4 ÷ 2 = 2, 4 ÷ 2 = 2. So 3 × 2 × 2 = 12 cubes. 96 counts 1 cm cubes, 48 divides 96 by 2 instead of by 8 (each 2 cm cube holds 8 small cubes), and 24 halves only two edges.
  26. 26. 30 L
    50 × 30 × 20 = 30 000 cm³, and 30 000 ÷ 1000 = 30 L. 3 L and 300 L shift the place value, and 100 L adds 50 + 30 + 20.
  27. 27. 4200 cm³
    20 × 7 = 140 for one layer, and 140 × 30 = 4200 cm³. 57 adds the edges, 140 is one layer, and 600 multiplies only 20 × 30.
  28. 28. 4 cm
    4 × 4 × 4 = 64. 8 cm would give 512 cm³ (8 × 8 = 64 is only one face), 16 cm divides 64 by 4, and 21 is about 64 ÷ 3.
  29. 29. 48 cm³
    Full block: 5 × 4 × 3 = 60 cm³. The hole is 2 × 2 × 3 = 12 cm³, because it goes through all 3 layers. 60 − 12 = 48 cm³. 56 removes only one layer of the hole, 60 removes nothing, and 12 is the hole itself.
  30. 30. Possible answers: 6 × 6 × 1, 4 × 3 × 3, 9 × 2 × 2, 12 × 3 × 1, 6 × 3 × 2, 18 × 2 × 1, 36 × 1 × 1. Any three whose product is 36 are correct.
    Check each box by multiplying: the three edges must multiply to 36. Boxes can look very different and still hold the same number of cubes.

Related practice

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