A cube of side 4 cm is cut into cubes of side 1 cm. What is the ratio of the surface areas of the original cube and all the cut-out cubes? (Note that there is no change in volume but a big change in the surface area. This property has major consequences in the biological world.)
Step-by-step solution
To find: Ratio of surface areas (original : all small cubes)
Idea: Volume stays the same when you cut, so count the small cubes by volume. Every cut makes new faces, so the total surface area grows.
- Surface area of original cube = 6 × 42 = 96 cm2.½ mark
- Number of small cubes = volume ÷ volume = 43 ÷ 13 = 64.1 mark
- Surface area of one small cube = 6 × 12 = 6 cm2; all 64 cubes: 64 × 6 = 384 cm2.½ mark
- Ratio = 96 : 384 = 1 : 4 (divide both by 96). Cutting makes the total surface 4 times as large; this is why small cells, which have much more surface for their volume, can take in food and oxygen quickly.1 mark
Check: Volumes: 64 cm3 before, 64 × 1 = 64 cm3 after ✓ (no change in volume).
Answer to write in the exam
SA of original cube = 6 × 42 = 96 cm2
Number of small cubes = 43 ÷ 13 = 64
Total SA of small cubes = 64 × 6 × 12 = 384 cm2
Ratio = 96 : 384
∴ Required ratio = 1 : 4
Common mistakes that cost marks
- Counting 16 small cubes (4 × 4) instead of 64 (4 × 4 × 4).
- Writing the ratio upside down (4 : 1); the original cube comes first.
- Assuming the surface area stays the same because the volume did.
How this can come in the exam
Assertion (A): When a cube of side 3 cm is cut into 1 cm cubes, the total surface area becomes 3 times as large.
Reason (R): Cutting a solid into pieces does not change its total volume.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
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(B) Both A and R are true, but R is not the correct explanation of A.
A: 54 cm2 becomes 27 × 6 = 162 cm2, 3 times, true. R is true, but equal volume does not explain the rise in area; the new faces made by cutting do.
A cube of side 6 cm is cut into cubes of side 2 cm. Find the ratio of the surface area of the big cube to the total surface area of the small cubes.
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Big: 6 × 36 = 216 cm2. Small cubes: (6 ÷ 2)3 = 27, each 24 cm2, total 648 cm2. Ratio 216 : 648 = 1 : 3.Try one yourself
A cube of side 10 cm is cut into cubes of side 2 cm. Find the ratio of surface areas, original : cut-out.
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600 : (125 × 24) = 600 : 3000 = 1 : 5.
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