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S4 Maths: Mensuration of 3D Solids (Paper 3 (challenge))
S4 Maths: Mensuration of 3D Solids (Paper 3 (challenge))
Paper 3 (challenge): 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 solid is enlarged 1.5 times and stays similar to the original. Find the factor by which the volume grows.
- FoundationA cone has radius 4 cm and slant height 9 cm. What is the angle of the sector in its net?
- FoundationA composite solid is a cylinder of radius 3 cm and height 8 cm with a hemisphere of the same radius on top. Find its total volume.
- IntermediateA pyramid has a rectangular base of 5 cm by 4 cm and volume 60 cubic cm. Find its height.
- IntermediateWhen the radius of a sphere doubles, by what factor do its surface area and volume grow?
- IntermediateA cube has edge 6 cm. Find its space diagonal, from one vertex to the opposite vertex.
- ChallengeA cone has volume 96 pi cubic cm and height 8 cm. Find its radius, slant height and curved surface area.
- ChallengeTwo similar cylinders have volumes in the ratio 8 : 125. Find the ratio of their surface areas.
- ChallengeA cylinder of radius 3 cm and height 4 cm is full of water and poured into a cone of radius 3 cm and height 12 cm. How many pours fill the cone?
- Error redo(Error clinic) A student uses the length ratio 3 : 5 directly as the volume ratio. What are the true area and volume ratios of the pair?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | Volume factor = 1.5 cubed = 3.375 times. | When every length grows by a factor k, the volume grows by k cubed. |
| 2 | Sector angle = 360 x 4 / 9, about 160 degrees. | The sector angle rises with the radius and falls as the slant height grows. |
| 3 | Cylinder 72 pi plus hemisphere 18 pi gives 90 pi, about 282.743 cubic cm. | Split a composite solid into parts, compute each volume and add them. |
| 4 | One third x 20 x h = 60, so h = 9 cm. | To recover the height, first divide the volume by the base area and undo the factor of one third. |
| 5 | Surface area grows by 2 squared = 4 times, from 100 to 400 square cm, and volume grows by 2 cubed = 8 times. | The radius is a length, so area scales with its square and volume with its cube. |
| 6 | The diagonal is 6 root 3, about 10.392 cm. | A space diagonal uses root three times the edge, while a face diagonal only uses root two. |
| 7 | r squared = 96 x 3 / 8 = 36, so r is 6 cm; slant height = the square root of 36 plus 64 = 10 cm; curved surface = 60 pi, about 188.496 square cm. | Recover the radius from the volume, take the root for the slant height, then substitute into pi r l. |
| 8 | Length ratio = cube root 8 : cube root 125 = 2 : 5, so surface area ratio = 4 : 25. | Take the cube root of the volume ratio to get the length ratio, then square it for the area ratio. |
| 9 | The cylinder holds 36 pi and the cone also holds 36 pi, so 1 pour fills it exactly. | Compute both volumes and divide rather than guessing from the heights. |
| 10 | Surface area ratio = 9 : 25 and volume ratio = 27 : 125, so neither equals 3 : 5. | The length ratio must be squared for areas and cubed for volumes. |
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 →
Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.
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