Worksheets · Higher

Area and volume in similar figures

8 exam-style questions, grades 6 to 8. Worked solutions and the marks are on the last page.

Question 1Non-calculator · 3 marks
Two cylinders are mathematically similar. The smaller has height 6 cm and volume $160\text{ cm}^3$ . The larger has height 9 cm.
(a) Work out the volume of the larger cylinder. (3)
Question 2Non-calculator · 1 mark
(a) Circle B has four times the radius of circle A. Which statement about their areas is correct? (1)
1. The area of B is 4 times the area of A.
2. The area of B is 8 times the area of A.
3. The area of B is 16 times the area of A.
4. The area of B is 64 times the area of A.
Question 3Non-calculator · 3 marks
(a) Two similar containers have volumes in the ratio 64 : 125. The smaller container is 12 cm high. Find the height of the larger container. (3)
Question 4Calculator · 3 marks
(a) Two similar solid models have surface areas in the ratio 9 : 25. The smaller model has volume 81 cm³. Find the volume of the larger model. (3)
Question 5Non-calculator · 4 marks
Two solid shapes are mathematically similar. Their surface areas are $72\text{ cm}^2$ and $162\text{ cm}^2$ . The smaller shape has volume $96\text{ cm}^3$ .
(a) Work out the volume of the larger shape. (4)
Question 6Calculator · 3 marks
A map has a scale of 1 : 25 000. A lake has an area of $12\text{ cm}^2$ on the map.
(a) Work out the real area of the lake in square kilometres. (3)
Question 7Calculator · 3 marks
(a) Two similar closed solid models are made from the same material. Their surface areas differ by 84 cm² and are in the ratio 9 : 16. The smaller model has volume 81 cm³. A third similar model has surface area equal to the sum of the first two surface areas. Find the volume of the third model. You must show your working. (3)
Question 8Non-calculator · 4 marks
A solid cone is cut by a plane parallel to its base, one third of the way down from the vertex (measured along the height). This cuts off a small cone and leaves a frustum.
(a) Show that the ratio of the volume of the small cone to the volume of the frustum is $1 : 26$ . (4)

Worked solutions and marks

Question 1

(a) $540\text{ cm}^3$

Length scale factor $k = \frac{9}{6} = 1.5$ .
Volumes scale by $k^3$ .
$1.5^3 = 3.375$
$160 \times 3.375 = 540$

P1 Length scale factor 1.5.
P1 Cubing it for volume.
A1 $540\text{ cm}^3$ .

Question 2

(a) The area of B is 16 times the area of A.

Circle area is $\pi r^{2}.$
Replacing r by 4r gives $\pi$ (4r) $^{2}$ = $16\pi r^{2}$ , so the area factor is 16.

B1 Correct answer: The area of B is 16 times the area of A.

Question 3

(a) $15$ cm

$(125/64)^{1/3}$
$12\times 5/4$
For similar solids, the volume ratio is the cube of the length ratio.
The length ratio is 4 : 5.
Larger height = 12 $\times$ 5/4 = 15 cm.

P1 Establishing $(125/64)^{1/3}$ or an equivalent valid method.
P1 Establishing $12\times 5/4$ or an equivalent valid method.
A1 Correct answer: $15$ cm

Question 4

(a) $375$ cm³

$\sqrt{25/9}$
$81\times (5/3)^{3}$
Take square roots of the area ratio to get the length ratio 3 : 5.
The volume scale factor is (5/3) $^{3}$ = 125/27.
81 $\times$ 125/27 = 375 cm³.

P1 Establishing $\sqrt{25/9}$ or an equivalent valid method.
P1 Establishing $81\times (5/3)^{3}$ or an equivalent valid method.
A1 Correct answer: $375$ cm³

Question 5

(a) $324\text{ cm}^3$

Area scale factor.
$k^2 = \frac{162}{72} = \frac{9}{4}$
Length scale factor.
$k = \frac{3}{2}$
Volume scale factor.
$k^3 = \frac{27}{8}$
$96 \times \frac{27}{8} = 324$

P1 Area factor $\frac{9}{4}$ .
P1 Square-rooting to the length factor $\frac{3}{2}$ .
P1 Cubing to the volume factor $\frac{27}{8}$ .
A1 $324\text{ cm}^3$ .

Question 6

(a) $0.75\text{ km}^2$

1 cm on the map is 25 000 cm $= 250$ m $= 0.25$ km.
So $1\text{ cm}^2$ on the map is $0.25^2\text{ km}^2$ .
$0.25^2 = 0.0625$
$12 \times 0.0625 = 0.75$

P1 A length conversion: 1 cm to 0.25 km (or 250 m).
P1 Squaring the length factor for area.
A1 $0.75\text{ km}^2$ .

Question 7

(a) $375$ cm³

$84/(16-9)$
$81\times (5/3)^{3}$
The difference is 7 area-parts, so one part is 12 cm². The first two areas are 108 and 192 cm².
The third area is 300 cm², which is 25/9 times the smaller area.
The length factor is 5/3, so the volume factor is 125/27.
Third volume = 81 $\times$ 125/27 = 375 cm³.

P1 Establishing $84/(16-9)$ or an equivalent valid method.
P1 Establishing $81\times (5/3)^{3}$ or an equivalent valid method.
A1 Correct answer: $375$ cm³

Question 8

(a) Small cone $= \frac{1}{27}V$ ; frustum $= \frac{26}{27}V$ .

The small cone is similar to the whole cone with length scale factor $\frac{1}{3}$ .
Volume scale factor.
$\left(\tfrac{1}{3}\right)^3 = \tfrac{1}{27}$
If the whole cone has volume $V$ , the small cone has $\frac{V}{27}$ and the frustum $V - \frac{V}{27} = \frac{26V}{27}$ .
Ratio $\frac{V}{27} : \frac{26V}{27} = 1 : 26$ .

P1 Length scale factor $\frac{1}{3}$ for the small cone.
P1 Volume factor $\frac{1}{27}$ .
P1 The frustum as $V - \frac{V}{27}$ .
A1 Reaching $1 : 26$ with the frustum expressed correctly.

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Area and volume in similar figures

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