IGCSE · Practice questions · Surface area and volume: prisms and cylinders

Area times length.

Six original Cambridge-style questions on the volume and surface area of cuboids, prisms and cylinders, including exact answers and working backwards.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Cross-section first, then the length.

Prism: \(V = Al\), \(A\) = cross-section area (given).
Cylinder: \(V = \pi r^2 h\) and curved surface \(2\pi rh\) (both given).
Total surface area: add every face; for a cylinder add \(2\pi r^2\) for the ends.
Open containers: leave out the missing face.
Backwards: substitute the volume and solve; round only at the end.
Capacity: \(1000\text{ cm}^3 = 1\) litre.

Recall

[2 marks]

A cuboid measures 7 cm by 4 cm by 2.5 cm. Find (a) its volume (b) its surface area.

(a) \(7 \times 4 \times 2.5 = 70\text{ cm}^3\) ✓
(b) \(2(7 \times 4 + 7 \times 2.5 + 4 \times 2.5) = 2(28 + 17.5 + 10) = 111\text{ cm}^2\) ✓

Calculation

[2 marks]

A cylinder has radius 3.5 cm and height 9 cm. Calculate (a) its volume (b) its curved surface area.

(a) \(\pi \times 3.5^2 \times 9 = 346.3... = 346\text{ cm}^3\) ✓
(b) \(2\pi \times 3.5 \times 9 = 197.9... = 198\text{ cm}^2\) ✓

Calculation

[3 marks]

A prism is 20 cm long. Its cross-section is a right-angled triangle with sides 5 cm, 12 cm and 13 cm.

Find (a) its volume (b) its total surface area.

Cross-section \(= \tfrac{1}{2} \times 5 \times 12 = 30\text{ cm}^2\). ✓
(a) \(V = 30 \times 20 = 600\text{ cm}^3\) ✓
(b) Ends \(2 \times 30 = 60\); rectangles \((5 + 12 + 13) \times 20 = 600\); total \(660\text{ cm}^2\) ✓

Calculation

[2 marks]

Without a calculator, find the volume and the total surface area of a closed cylinder with radius 5 cm and height 8 cm. Give your answers in terms of \(\pi\).

\(V = \pi \times 25 \times 8 = 200\pi\text{ cm}^3\) ✓
Surface area \(= 2\pi \times 5 \times 8 + 2 \times 25\pi = 80\pi + 50\pi = 130\pi\text{ cm}^2\) ✓

Analysis

[3 marks]

A cylindrical jug is 15 cm tall and holds exactly 1 litre when full.

Calculate the radius of the jug.

1 litre \(= 1000\text{ cm}^3\). ✓
\(\pi r^2 \times 15 = 1000\), so \(r^2 = \dfrac{1000}{15\pi} = 21.22...\) ✓
\(r = 4.606... = 4.61\) cm ✓

Analysis

[4 marks]

A water trough is a prism 2 m long. Its cross-section is a semicircle of radius 30 cm.

(a) Calculate the capacity of the trough in litres.

(b) Water flows into the empty trough at 8 litres per minute. How long does it take to fill?

Cross-section \(= \tfrac{1}{2} \times \pi \times 30^2 = 450\pi = 1413.7...\text{ cm}^2\). ✓
Length 2 m = 200 cm, so \(V = 1413.7... \times 200 = 282\,743...\text{ cm}^3\). ✓
(a) \(282\,743... \div 1000 = 283\) litres ✓
(b) \(282.74... \div 8 = 35.3\) minutes ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.