TACHS: Packing cubes: count by edges, then multiply
Count how many cubes fit in a box by counting along each edge, handle leftover space, and work backward to a box size.
How do you count how many small cubes fit inside a box?
Short answer
Divide each box edge by the cube's side and drop any remainder. Multiply the three counts: cubes along the length × along the width × the number of layers.
Know first: Volume of a box: length × width × height, Dividing with a remainder
Step 1 of 6
Fill the box one layer at a time
Imagine filling a box with cubes. A whole number of cubes fits along the length and another along the width. Together they make one layer. Then count how many layers stack up the height. Multiply the three counts.
When every edge is an exact multiple of the cube's side, this count equals box volume ÷ cube volume. When an edge leaves a gap, only whole cubes count, so you drop the remainder on each edge before you multiply.
| Step | Box 20 by 15 by 10 cm, 5 cm cubes |
|---|---|
| Along the length | 20 ÷ 5 = 4 |
| Along the width | 15 ÷ 5 = 3 |
| Layers up the height | 10 ÷ 5 = 2 |
| Multiply | 4 × 3 × 2 = 24 cubes |
Divide each edge by the cube's side length, never by the cube's volume.
Step 2 of 6
Edge counts, then a volume check
Worked example: A box of 5 cm cubes
A box has inside dimensions 20 cm by 15 cm by 10 cm. Cubes with 5 cm sides are packed in rows parallel to the edges. How many cubes fit?
- A9
- B24
- C125
- D3,000
- 1
Count along each edge
20 ÷ 5 = 4, 15 ÷ 5 = 3 and 10 ÷ 5 = 2.
- 2
Multiply
4 × 3 × 2 = 24 cubes.
- 3
Check with volume
Box: 20 × 15 × 10 = 3,000 cubic cm. Cube: 5 × 5 × 5 = 125 cubic cm. 3,000 ÷ 125 = 24. The two methods agree because every edge divides evenly.
- 4
See why the other choices are there
9 adds the edge counts instead of multiplying them. 125 is the volume of one cube. 3,000 is the volume of the box.
Answer
24 cubes fit.
Wrong: A box 11 by 9 by 7 cm holds 2 cm cubes: 693 ÷ 8 is about 86 cubes.
Right: Edge counts: 11 ÷ 2 → 5, 9 ÷ 2 → 4, 7 ÷ 2 → 3. Cubes: 5 × 4 × 3 = 60.
Each edge leaves a 1 cm sliver that cannot hold a whole cube. Dividing volumes counts those slivers as if they could be filled.
Step 3 of 6
Work backward from a count
Some questions give the number of cubes and ask for a box edge. A box holds 30 cubes with 2 cm sides, and its inside is 10 cm long and 6 cm wide. One layer holds 10 ÷ 2 = 5 by 6 ÷ 2 = 3, or 15 cubes. So there are 30 ÷ 15 = 2 layers, and the height is 2 × 2 = 4 cm.
If you know the counts along each edge, multiply each by the cube's side to get the box edges. With 4, 3 and 2 cubes of side 3 cm, the box is 12 by 9 by 6 cm, for a volume of 648 cubic cm.
Step 4 of 6
Count whole cubes along each edge
Check yourself · Question 1
A box has inside dimensions 20 cm by 12 cm by 8 cm. Cubes with 4 cm sides are packed in rows with no gaps. How many cubes fit?
Answer: B
Edge counts: 20 ÷ 4 = 5, 12 ÷ 4 = 3 and 8 ÷ 4 = 2. Cubes: 5 × 3 × 2 = 30.
Trap answer: 1,920
Box volume instead of a count
- Why it looks right
- Multiplying the three box edges is the first familiar move, and it gives a big, confident number.
- Why it's wrong
- 1,920 measures space in cubic centimeters. Each 4 cm cube takes up 64 of them, so the count is much smaller.
- The right answer
- 30 cubes: 5 × 3 × 2.
Check yourself · Question 2
A box measures 14 cm by 11 cm by 9 cm inside. Cubes with 3 cm sides are packed in rows parallel to the edges, and unused space is allowed. What is the greatest number of whole cubes that fit?
Answer: A
Edge counts: 14 ÷ 3 → 4, 11 ÷ 3 → 3 and 9 ÷ 3 = 3. Cubes: 4 × 3 × 3 = 36.
Check yourself · Question 3
A box measuring 12 cm by 6 cm by 6 cm is filled with 2 cm cubes. If 3 cm cubes are used instead, how many fewer cubes fit?
Answer: B
2 cm cubes: 6 × 3 × 3 = 54. 3 cm cubes: 4 × 2 × 2 = 16. Difference: 54 − 16 = 38.
Common mistake
"I found the box volume and divided by the cube volume."
- Why it's tempting
- It works whenever the edges divide evenly, so it feels like a rule.
- Do this instead
- Count along each edge first and drop remainders. Use the volume method only as a check, when every edge divides exactly.
Step 5 of 6
What to remember
Remember
- Divide each box edge by the cube's side length, and drop any remainder.
- Multiply the three counts: length × width × layers.
- Box volume ÷ cube volume matches only when every edge divides evenly; work backward by dividing the count by one layer.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, how many cubes fit along each edge, and which wrong choice is the volume of the box?