Week 27 · Same volume, different shape; the volume formula · 3 lessons
1
Same volume, different shape
Discover Mathematics P5: Term 2, Unit 11, Lessons 4–6
GoalSee that boxes with different shapes can have the same volume, and find boxes with a given volume.
New words
dimensions · the length, width and height of a solidrearrange · to put the same cubes together in a different wayequal volumes · two solids that take up the same amount of space
Explanation
Take 24 unit cubes. You can build a box 4 long, 3 wide and 2 high, or a flat box 6 by 4 by 1, or a tall tower 2 by 2 by 6. The boxes look very different, but they use the same 24 cubes, so they all have the same volume: 24 cubic units. Volume does not depend on the shape. It depends only on how many unit cubes fill the solid. To find out whether two boxes have the same volume, work out the volume of each by layers (length × width × height) and compare. To find boxes with a given volume, look for three numbers that multiply to give it. For 24 the triples are 4 × 3 × 2, 6 × 4 × 1, 8 × 3 × 1, 12 × 2 × 1 and 2 × 2 × 6.
Length
Width
Height
Volume
4
3
2
24
6
4
1
24
8
3
1
24
12
2
1
24
2
2
6
24
Worked examples
Box A is 6 cm × 4 cm × 3 cm. Box B is 8 cm × 3 cm × 3 cm. Volume of A: 6 × 4 = 24, 24 × 3 = 72 cm³. Volume of B: 8 × 3 = 24, 24 × 3 = 72 cm³. The two boxes have the same volume.
Find boxes made of exactly 30 unit cubes. The numbers must multiply to 30: 5 × 3 × 2, 10 × 3 × 1, 6 × 5 × 1, 15 × 2 × 1 and 30 × 1 × 1 all work.
Karim builds a tower of 36 cubes that is 12 long, 3 wide and 1 high. He rearranges the cubes into a box with a base of 4 by 3. It is 36 ÷ 12 = 3 layers high: 4 × 3 × 3 = 36. The volume has not changed.
Class activity
Twelve-cube challenge. Use 12 sugar cubes or blocks. Build every box you can that uses all 12 cubes, and write its length, width and height. How many different boxes did you find? Then repeat with 16 cubes. Which number of cubes gives more different boxes?
Practice
1
Do a box 6 × 2 × 3 and a box 4 × 3 × 3 have the same volume?
6 × 2 × 3 = 36 and 4 × 3 × 3 = 36. Yes, both are 36 cubic units.
2
Give three different boxes whose volume is 20 cm³.
For example 10 × 2 × 1, 5 × 4 × 1 and 5 × 2 × 2.
3
Box A is 5 × 4 × 2 and box B is 10 × 2 × 2. Which has the greater volume?
A: 5 × 4 × 2 = 40. B: 10 × 2 × 2 = 40. They are equal.
4
Which has the greater volume, a box 9 × 3 × 2 or a box 6 × 4 × 3, and by how much?
9 × 3 × 2 = 54 and 6 × 4 × 3 = 72. The second is greater by 72 − 54 = 18 cubic units.
5
48 cubes make a flat box 8 by 6 by 1. How could you rearrange them into a box with a base of 4 by 3? How tall will it be?
A base of 4 × 3 holds 12 cubes per layer, and 48 ÷ 12 = 4, so it is 4 layers (4 high): 4 × 3 × 4.
