Similar shapes: length, area & volume ratios – IGCSE Maths exam questions with mark scheme

1Grade 4–5Calculator3 marks

Triangle $ABC$ is mathematically similar to triangle $DEF$.

Angle $ABC$ = angle $DEF$.

$AB = 6$ cm, $BC = 8$ cm, $DE = 9$ cm and $DF = 15$ cm.

(a) Work out the length of $EF$. [2]

(b) Work out the length of $AC$. [1]

(a) for scale factor $\dfrac{9}{6}$ or $1.5$ oe, or $\dfrac{EF}{8} = \dfrac{9}{6}$M1
$12$ (cm)A1
(b) $10$ (cm) (ft $15 \div$ their scale factor)B1
Answer: (a) 12 cm (b) 10 cm
Examiner tip: Match corresponding sides carefully: AB ↔ DE, BC ↔ EF, AC ↔ DF — then multiply to go from small to large and divide to go back.

2Grade 6–7HigherCalculator5 marks

Two cylinders, $\mathbf{P}$ and $\mathbf{Q}$, are mathematically similar.

Cylinder $\mathbf{P}$ has height 6 cm. Cylinder $\mathbf{Q}$ has height 9 cm.

The total surface area of cylinder $\mathbf{P}$ is 120 cm$^2$.

(a) Work out the total surface area of cylinder $\mathbf{Q}$. [2]

The volume of cylinder $\mathbf{Q}$ is 810 cm$^3$.

(b) Work out the volume of cylinder $\mathbf{P}$. [3]

(a) for $\left(\dfrac{9}{6}\right)^2$ or $1.5^2$ or $2.25$ oeM1
$270$ (cm$^2$)A1
(b) for $\left(\dfrac{9}{6}\right)^3$ or $1.5^3$ or $3.375$ oe, or $\left(\dfrac{6}{9}\right)^3$ or $\dfrac{8}{27}$M1
for $810 \div 3.375$ or $810 \times \dfrac{8}{27}$M1
$240$ (cm$^3$)A1
Answer: (a) 270 cm² (b) 240 cm³
Examiner tip: Area scales by the square and volume by the cube of the length scale factor — using 1.5 for the volume loses both method marks.

3Grade 6–7HigherCalculator4 marks

Two statues are mathematically similar and are made from the same material.

The small statue has a height of 12 cm and a mass of 160 g.

The large statue has a height of 30 cm.

(a) Work out the mass of the large statue. [2]

It takes 75 ml of paint to cover the surface of the large statue.

(b) Work out how much paint is needed to cover the surface of the small statue. [2]

(a) for $\left(\dfrac{30}{12}\right)^3$ or $2.5^3$ or $15.625$ oeM1
$2500$ (g) (accept 2.5 kg)A1
(b) for $\left(\dfrac{30}{12}\right)^2$ or $2.5^2$ or $6.25$ oeM1
$12$ (ml)A1
Answer: (a) 2500 g (b) 12 ml
Examiner tip: Mass depends on volume (scale factor cubed) but paint depends on surface area (scale factor squared).

4Grade 8–9HigherCalculator4 marks

Two containers, $\mathbf{A}$ and $\mathbf{B}$, are mathematically similar.

The volume of container $\mathbf{A}$ is 400 cm$^3$. The volume of container $\mathbf{B}$ is 1350 cm$^3$.

The surface area of container $\mathbf{A}$ is 220 cm$^2$.

Work out the surface area of container $\mathbf{B}$. [4]

for $\dfrac{1350}{400}$ or $3.375$ or $\dfrac{27}{8}$ oe (volume scale factor)M1
for $\sqrt[3]{3.375} = 1.5$ oe (length scale factor)M1
for $220 \times 1.5^2$ oeM1
$495$ (cm$^2$)A1
Answer: 495 cm²
Examiner tip: You cannot go straight from a volume ratio to an area ratio — find the length scale factor (cube root) first, then square it.

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