Surface Area and Volume of Solids

14 Grade 10 Core Mathematics questions on Surface Area and Volume of Solids, with worked marking schemes. 14 questions are available on this topic in full.

1. Question 1

   5 marks

   A grain silo consists of a cylinder of base radius 2.1 m and height 5 m, with a hemisphere of the same radius fixed on top. Taking $\pi =\frac{22}{7}$, calculate the total volume of the silo.

   (5 marks)

   Marking scheme

   Award 1 mark per point, maximum 5. 1. Volume of cylinder = $\pi r^{2}h=\frac{22}{7}\times 2.1^2\times 5$. ✓ 2. = 69.3 m³. ✓ 3. Volume of hemisphere = $\frac{2}{3}\times \pi$r³ = $\frac{2}{3}\times \frac{22}{7}\times 2.1^3$. ✓ 4. = 19.40 m³. ✓ 5. Total volume = $69.3+19.40=88.70m^3$. ✓ Allow 88.7 m³ or 88.71 m³ from correct working.

2. Question 2

   1 mark

   A closed conical hopper has a base radius of 7 cm, a vertical height of 24 cm and a slant height of 25 cm. Taking $\pi =\frac{22}{7}$, its total surface area is:

   A. 154 cm²

   B. 704 cm²

   C. 528 cm²

   D. 550 cm²

   (1 mark)

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3. Question 3

   4 marks

   A solid metal sphere of radius 3 cm is melted down and recast, without waste, into a solid cylinder of base radius 2 cm. Taking π = 3.142, calculate the height of the cylinder.

   (4 marks)

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4. Question 4

   1 mark

   A spherical ball used in a school game has a radius of 10.5 cm. Taking $\pi =\frac{22}{7}$, the area of leather needed to cover its outer surface is:

   A. 1 386.0 cm²

   B. 346.5 cm²

   C. 693.0 cm²

   D. 4 851.0 cm²

   (1 mark)

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5. Question 5

   1 mark

   A conical paper cup used at a juice kiosk has a base radius of 3.5 cm and a vertical height of 12 cm. Taking $\pi =\frac{22}{7}$, its volume is:

   A. 616 cm³

   B. 154 cm³

   C. 462 cm³

   D. 231 cm³

   (1 mark)

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6. Question 6

   2 marks

   A water tank holds 7.5 m³ when full. Explain why its capacity is 7 500 litres.

   (2 marks)

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7. Question 7

   1 mark

   A concrete wheelchair ramp is a prism whose uniform cross-section is a right-angled triangle of base 1.2 m and height 0.5 m. The ramp is 3 m long. Its volume is:

   A. 0.9 m³

   B. 1.8 m³

   C. 0.3 m³

   D. 0.6 m³

   (1 mark)

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8. Question 8

   3 marks

   A cylindrical tank on a school compound holds 3.08 m³ of water when full. Its base radius is 0.7 m. Taking $\pi =\frac{22}{7}$, calculate the height of the tank.

   (3 marks)

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9. Question 9

   4 marks

   A camping tent is a prism 2.4 m long. Its cross-section is an isosceles triangle with a base of 1.6 m and two equal sloping sides of 1 m each. The tent has no canvas floor. Calculate (a) the vertical height of the cross-section, (b) the total area of canvas used for the two sloping faces and the two triangular ends.

   (4 marks)

   Marking scheme

   Award 1 mark per point, maximum 4. (a) 1. By Pythagoras, height = $\sqrt{1^2-0.8^2}=\sqrt{0.36}=0.6m$. ✓ (b) 2. Two sloping faces = $2\times (1\times 2.4)=4.8m^2$. ✓ 3. Two triangular ends = $2\times (\frac{1}{2}\times 1.6\times 0.6)=0.96m^2$. ✓ 4. Total canvas = $4.8+0.96=5.76m^2$. ✓ A candidate who includes a floor of $1.6\times 2.4$ forfeits mark 4.

10. Question 10

    1 mark

    A closed cylindrical tin of base radius 7 cm and height 10 cm is to be painted on the outside, including the top and the bottom. Taking $\pi =\frac{22}{7}$, the area painted is:

    A. 440 cm²

    B. 1 540 cm²

    C. 594 cm²

    D. 748 cm²

    (1 mark)

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11. Question 11

    1 mark

    A cylindrical water tank standing on a farm has a base radius of 1.4 m and a height of 2 m. Taking $\pi =\frac{22}{7}$, its volume is:

    A. 17.60 m³

    B. 3.08 m³

    C. 6.16 m³

    D. 12.32 m³

    (1 mark)

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12. Question 12

    4 marks

    A metal drum with an open top is used to collect rain water from a roof. It has a base radius of 0.35 m and a height of 0.9 m. Taking $\pi =\frac{22}{7}$, calculate the area of metal sheet used to make the drum.

    (4 marks)

    Marking scheme

    Award 1 mark per point, maximum 4. 1. Curved surface area = $2\pi rh=2\times \frac{22}{7}\times 0.35\times 0.9$. ✓ 2. = 1.98 m². ✓ 3. Base only (no lid, so one circle) = $\frac{22}{7}\times 0.35^2=0.385m^2$. ✓ 4. Total metal = $1.98+0.385=2.365m^2$. ✓ A candidate who adds two circular ends scores marks 1 and 2 only.

13. Question 13

    4 marks

    A solid concrete block used as a gate weight is a pyramid on a square base of edge 30 cm with a vertical height of 40 cm.

    (a) Calculate its volume.

    (b) Given that the concrete has a density of 2.4 g/cm³, determine the mass of the block in kilograms.

    (4 marks)

    Marking scheme

    Award 1 mark per point, maximum 4. (a) 1. Volume = $\frac{1}{3}$ × base area × height = $\frac{1}{3}\times (30\times 30)\times 40$. ✓ 2. = 12 000 cm³. ✓ (b) 3. Mass = volume × density = $12000\times 2.4=28800$ g. ✓ 4. = 28.8 kg. ✓ A candidate who omits the $\frac{1}{3}$ forfeits marks 1 and 2 but may earn method marks in (b).

14. Question 14

    1 mark

    A hemispherical bowl of internal radius 7 cm is filled to the brim with milk. Taking $\pi =\frac{22}{7}$, the volume of milk in the bowl is:

    A. 359.3 cm³

    B. 1 078.0 cm³

    C. 1 437.3 cm³

    D. 718.7 cm³

    (1 mark)

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