Free Surface area & volume questions for IGCSE Mathematics (0580), part of Mensuration. Each question has the answer and the marking points an examiner would look for.
Generate a full Surface Area & Volume worksheet →All Mensuration questionsA cylindrical tank of radius 10 cm contains water to a depth of 12 cm.
A solid metal sphere of radius 6 cm is placed in the tank and is completely covered by the water.
Calculate the new depth of the water.
[The volume, V, of a sphere with radius r is V = 43πr3.]
| Part | Answer and marking points | Marks |
|---|---|---|
| 14.9 cm sphere volume = 43π × 63 M1 rise = sphere volume ÷ (π × 102) M1 rise = 2.88 A1 14.9 A1 | 4 |
A cylindrical tank has internal radius 6 cm and stands on a horizontal floor. It contains water to a depth of h cm.
A solid ball of radius r cm, where 0 < r ≤ 6, is placed in the tank and rests on the bottom. The water level rises and the ball is then exactly covered: the water surface just touches the top of the ball. No water is lost.
[Volume of a sphere = 43πr3]
For some values of h, only one size of ball can be exactly covered.
Find the set of these values of h. Give any boundary values exactly.
A ball of radius 3 cm is exactly covered. The ball is removed, the water level returns to its original depth, and a ball of a different radius is placed in the tank. This ball is also exactly covered.
Find the radius of the second ball.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 0 < h < 4 or h = 4√2 water + ball fill a cylinder of height 2r: 36π × 2r = 36πh + 43πr3 M1 h = 2r − r327 A1 dhdr = 2 − r29 = 0 M1 r = 3√2 (≤ 6), maximum h = 6√2 − 2√2 = 4√2 A1 at r = 6 (largest ball that fits), h = 12 − 8 = 4 B1 h rises from 0 to 4√2 for 0 < r ≤ 3√2 and falls from 4√2 to 4 for 3√2 ≤ r ≤ 6 M1 so 4 ≤ h < 4√2 gives two radii (h = 4: r = 6 and r = 3√3 − 3) M1 0 < h < 4 A1 or h = 4√2 A1 | 9 |
| (b) | 5.37 cm (exactly 3√21 − 32) h = 6 − 1 = 5 B1 2r − r327 = 5, i.e. r3 − 54r + 135 = 0 M1 (r − 3)(r2 + 3r − 45) = 0, or trial / calculator, rejecting r = 3 M1 5.37 A1 | 4 |
The diagram shows a solid cylinder.
Calculate the volume of the cylinder.
Calculate the total surface area of the cylinder.
The cylinder is made from metal of density 8.9 g/cm³. Calculate the mass of the cylinder in kilograms.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 2830 cm³ π × 62 × 25 M1 2830 A1 | 2 |
| (b) | 1170 cm² 2 × π × 62 M1 2 × π × 6 × 25 M1 1170 A1 | 3 |
| (c) | see working their volume × density M1 ÷ 1000, correct answer A1 | 2 |
A prism has a cross-section that is a right-angled triangle with base 7 cm and height 3 cm. The prism is 13 cm long. Find its volume.
| Part | Answer and marking points | Marks |
|---|---|---|
| 136.5 cm³ ½ × 7 × 3 × 13 M1 136.5 A1 | 2 |
A solid metal sphere of radius 9 cm is melted and recast into a cylinder of radius 9 cm. Find the height of the cylinder.
| Part | Answer and marking points | Marks |
|---|---|---|
| 12 cm 43π × 93 M1 ÷ π × 92 M1 12 A1 | 3 |
A cylinder has radius 4 cm and height 12 cm.
Find its volume.
Find its total surface area.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 603 cm³ π × 42 × 12 M1 603 A1 | 2 |
| (b) | 402 cm² 2πr2 or 2πrh M1 both added M1 402 A1 | 3 |
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