IGCSE · Practice questions · Areas and volumes of similar shapes

k, k², k³.

Extended. Six original Cambridge-style questions on areas and volumes of similar shapes, ending with a best-value problem.

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

Get the length factor first.

Identify what you are given: two lengths, two areas or two volumes.
Reduce to $k$: divide, then square root (areas) or cube root (volumes).
Build up: $k$ for lengths, $k^2$ for areas, $k^3$ for volumes, capacity and mass.
Check direction: bigger shape, bigger answer.
Units: 1 m² = 10 000 cm²; 1 km² = 10¹⁰ cm².

Recall

[2 marks]

Two similar solids have lengths in the ratio $3 : 4$. Write down the ratio of (a) their surface areas (b) their volumes.

(a) $3^2 : 4^2 = 9 : 16$ ✓
(b) $3^3 : 4^3 = 27 : 64$ ✓

Calculation

[2 marks]

Two similar triangles have bases 6 cm and 15 cm. The smaller triangle has area 20 cm². Find the area of the larger triangle.

$k = 15 \div 6 = 2.5$; area factor $= 6.25$. ✓
$20 \times 6.25 = 125$ cm². ✓

Calculation

[3 marks]

Two similar jugs hold 250 ml and 2 litres. The smaller jug is 12 cm tall. How tall is the larger jug?

2 litres = 2000 ml, so the volume factor is $2000 \div 250 = 8$. ✓
$k = \sqrt[3]{8} = 2$. ✓
Height $= 12 \times 2 = 24$ cm. ✓

Calculation

[3 marks]

Two similar solids have surface areas 50 cm² and 72 cm². The smaller has volume 125 cm³. Find the volume of the larger.

Area factor $= 72 \div 50 = 1.44$, so $k = \sqrt{1.44} = 1.2$. ✓
Volume factor $= 1.2^3 = 1.728$. ✓
$125 \times 1.728 = 216$ cm³. ✓

Analysis

[3 marks]

A map has a scale of 1 : 50 000. A lake has an area of 6 cm² on the map. Find the real area of the lake in km².

Area factor $= 50\,000^2 = 2.5 \times 10^{9}$. ✓
Real area $= 6 \times 2.5 \times 10^{9} = 1.5 \times 10^{10}$ cm². ✓
$1$ km $= 10^{5}$ cm, so $1$ km² $= 10^{10}$ cm². Area $= 1.5$ km². ✓

Analysis

[5 marks]

Two paint tins are similar. The small tin is 10 cm tall and holds 1.25 litres; its label has area 150 cm². The large tin is 16 cm tall.

(a) Find the capacity of the large tin. (b) Find the area of the large tin's label. (c) The small tin costs $4.50 and the large tin costs $16.00. Which is better value? Show your working.

$k = 16 \div 10 = 1.6$. ✓
(a) $1.25 \times 1.6^3 = 1.25 \times 4.096 = 5.12$ litres. ✓
(b) $150 \times 1.6^2 = 150 \times 2.56 = 384$ cm². ✓
(c) Small: $4.50 \div 1.25 = 3.60$, so $3.60 per litre. Large: $16.00 \div 5.12 = 3.125$, about $3.13 per litre. ✓
The large tin is better value. ✓

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.