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S4 Maths: Mensuration of 3D Solids (Paper 2 (intermediate))
S4 Maths: Mensuration of 3D Solids (Paper 2 (intermediate))
Paper 2 (intermediate): ten questions on mensuration of 3D solids with answers and one-line notes.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 |
| Suggested time | 20 |
| Difficulty mix | Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1 |
| Chapter | Chapter: S4 Maths: Mensuration of 3D Solids |
| Level | S4 |
02 Questions (10 items)
- FoundationA sphere has radius 3 cm. Find its surface area in terms of pi and to three decimals.
- FoundationA cylinder has radius 7 cm and height 2 cm. Find its curved surface area, excluding the two bases.
- FoundationA pyramid has a square base of side 6 cm and height 5 cm. Find its volume.
- IntermediateA cone has radius 5 cm and slant height 13 cm. Find its height and volume.
- IntermediateA hemisphere has radius 6 cm. Find its volume in terms of pi and to three decimals.
- IntermediateA cylindrical pipe has inner radius 2 cm and length 50 cm. Find the volume of the space inside it.
- ChallengeTwo similar cylinders have surface areas in the ratio 9 : 25. Find the ratio of their lengths and of their volumes.
- ChallengeA cone has radius 3 cm and slant height 5 cm. Find its total surface area.
- ChallengeA sphere has volume 36 pi cubic cm. Find its radius and surface area.
- Error redo(Error clinic) A student thinks a cylinder and a cone with the same base and height have equal volume. Find both volumes for radius 3 cm and height 6 cm.
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | Surface area = 4 x pi x 9 = 36 pi, about 113.097 square cm. | Surface area is four pi r squared, so the power is two rather than three. |
| 2 | Curved surface = 2 x pi x 7 x 2 = 28 pi, about 87.965 square cm. | The curved surface is only the side band, equal to the circumference of the base times the height. |
| 3 | Volume = one third x 36 x 5 = 60 cubic cm. | A pyramid takes one third of base area times height, so compute the base area first. |
| 4 | Height = the square root of 169 minus 25 = 12 cm, and volume = one third x pi x 25 x 12 = 100 pi, about 314.159 cubic cm. | Subtract the squares to recover the missing leg, then use the height for the volume, never the slant height. |
| 5 | Volume = two thirds x pi x 216 = 144 pi, about 452.389 cubic cm. | A hemisphere is half a sphere, so the factor four thirds becomes two thirds. |
| 6 | Volume = pi x 4 x 50 = 200 pi, about 628.319 cubic cm. | The pipe is a cylinder, and the question asks for the space inside, so the usual cylinder formula applies. |
| 7 | Length ratio = root 9 : root 25 = 3 : 5, and volume ratio = 27 : 125. | Take the square root of the area ratio to reach the length ratio, then cube it for the volume ratio. |
| 8 | Total surface = pi x 3 x 5 plus pi x 9 = 15 pi + 9 pi = 24 pi, about 75.398 square cm. | A cone has one circular base, so the total is the curved surface plus a single circle. |
| 9 | Four thirds pi r cubed equals 36 pi, so r cubed is 27 and r is 3 cm; surface area = 4 pi x 9 = 36 pi, about 113.097 square cm. | Cancel pi first, take the cube root to get the radius, then put it into the surface area formula. |
| 10 | Cylinder 54 pi, about 169.646 cubic cm, and cone 18 pi, about 56.549 cubic cm, so the cone is one third of the cylinder. | The cone carries an extra factor of one third, so the two volumes differ by a factor of three. |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
For any mensuration question, sketch the solid first and label the radius, the height and the slant height on the diagram. The volume of a cone or pyramid is always one third of the prism with the same base and the same height, so the factor of one third must never be dropped. The volume of a sphere is four thirds pi r cubed while its surface area is four pi r squared, so the two formulas differ only in the power and the coefficient and must be kept apart. The net of a cone is a sector whose arc equals the circumference of the base, so its angle is 360 degrees times the radius divided by the slant height. When two solids are similar, the ratio of areas is the square of the ratio of lengths and the ratio of volumes is the cube, so find the length ratio first and then square or cube it to avoid swapping the two.Tutor Registration →
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