Lengths, areas and volumes of similar figures

25 min

Twice as tall, eight times as much water

A school collects rainwater from its roof in a plastic tank. It plans to add a second tank of exactly the same shape, but 1.5 times as tall. Will it hold 1.5 times as much water? No: it is also 1.5 times as wide and 1.5 times as deep, so it holds $1.5 \times 1.5 \times 1.5 \approx 3.4$ times as much. Lengths, areas and volumes of similar figures grow at three different rates.

Definition

If two figures are similar with length scale factor $k$ (every length of the image is $k$ times the matching length of the object), then

the area scale factor is $k^{2}$ : every area (and surface area) is multiplied by $k^{2}$ ;
the volume scale factor is $k^{3}$ : every volume (and capacity) is multiplied by $k^{3}$ .

Example: a scale factor of 3 multiplies lengths by 3, areas by 9 and volumes by 27.

Important

Why $k^{2}$ and $k^{3}$ . A unit square becomes a $k \times k$ square, of area $k^{2}$ ; any flat shape can be filled with tiny squares, and each is multiplied by $k^{2}$ . A unit cube becomes a $k \times k \times k$ cube, of volume $k^{3}$ , and any solid can be filled with tiny cubes.

	Object : image	Example, $k = 2$
Lengths	$1 : k$	$1 : 2$
Areas	$1 : k^{2}$	$1 : 4$
Volumes	$1 : k^{3}$	$1 : 8$

For an enlargement with a negative scale factor, use $∣ k ∣$ : with $k = - 2$ , areas are multiplied by 4.

Worked example 1: area on a map

On a map of scale $1 : 50\,000$ , a farm covers $3\ \mathrm{cm^2}$ . What is its real area in hectares ( $1\ \text{ha} = 10\,000\ \mathrm{m^2}$ )?

Solution The length scale factor is $k = 50\,000$ , so the area scale factor is $k^2 = 2.5 \times 10^9$ .

Real area $= 3 \times 2.5 \times 10^9 = 7.5 \times 10^9\ \mathrm{cm^2}$ . Since $1\ \mathrm{m^2} = 10^4\ \mathrm{cm^2}$ , this is $7.5 \times 10^5\ \mathrm{m^2} = 75$ ha.

Worked example 2: the two tanks

The small tank is 1.2 m tall and holds 1000 litres; 2 litres of paint cover its outside. The similar large tank is 1.8 m tall. How much does it hold, and how much paint does it need?

Solution $k = \frac{1.8}{1.2} = 1.5$ .

Capacity (a volume): $1000 \times 1.5^3 = 1000 \times 3.375 = 3375$ litres.

Paint (covers an area): $2 \times 1.5^2 = 2 \times 2.25 = 4.5$ litres.

Worked example 3: from areas back to lengths

Two similar triangles have areas $18\ \mathrm{cm^2}$ and $50\ \mathrm{cm^2}$ . A side of the smaller one is 6 cm. Find the matching side of the larger one.

Solution $k^2 = \frac{50}{18} = \frac{25}{9}$ , so $k = \sqrt{\frac{25}{9}} = \frac{5}{3}$ . The side is $6 \times \frac{5}{3} = 10$ cm.

Worked example 4: from volumes to areas

Two similar solids have volumes $64\ \mathrm{cm^3}$ and $216\ \mathrm{cm^3}$ . The smaller has surface area $80\ \mathrm{cm^2}$ . Find the surface area of the larger.

Solution $k^3 = \frac{216}{64} = \frac{27}{8}$ , so $k = \sqrt[3]{\frac{27}{8}} = \frac{3}{2}$ and $k^2 = \frac{9}{4}$ . Surface area $= 80 \times \frac{9}{4} = 180\ \mathrm{cm^2}$ .

Worked example 5: a model car

A model car is made to a scale of $1 : 18$ . The real car is 4.5 m long and its windscreen has an area of $1.35\ \mathrm{m^2}$ . Find the length of the model and the area of its windscreen in $\mathrm{cm^2}$ .

Solution Length: $\frac{450}{18} = 25$ cm. Area: $\frac{1.35}{18^2} = \frac{1.35}{324} \ \mathrm{m^2} = \frac{13\,500}{324}\ \mathrm{cm^2} \approx 41.7\ \mathrm{cm^2}$ .

Warning

Do not use $k$ for an area or a volume. "Twice as tall" does not mean "twice as much paint" (it is 4 times) or "twice as much water" (8 times).

Convert area and volume units with the same powers: $1\ \mathrm{m} = 100\ \mathrm{cm}$ , so $1\ \mathrm{m^2} = 10\,000\ \mathrm{cm^2}$ and $1\ \mathrm{m^3} = 1\,000\,000\ \mathrm{cm^3}$ .

Tip

Always find the length scale factor $k$ first. Given an area ratio, take its square root; given a volume ratio, take its cube root. Then square or cube $k$ for what you need.

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Practice

The last two are harder.

Two similar rectangles have length scale factor 3. The smaller has area $5\ \mathrm{cm^2}$ . Find the area of the larger.
Matching sides of two similar triangles are 4 cm and 10 cm. The smaller has area $12\ \mathrm{cm^2}$ . Find the area of the larger.
On a map of scale $1 : 25\,000$ , a lake covers $8\ \mathrm{cm^2}$ . Find its real area in hectares.
Two similar jugs are 12 cm and 18 cm tall. The smaller holds 0.8 litres. How much does the larger hold?
Two similar shapes have areas $16\ \mathrm{cm^2}$ and $36\ \mathrm{cm^2}$ . The smaller has perimeter 20 cm. Find the perimeter of the larger.
Two similar solids have volumes $27\ \mathrm{cm^3}$ and $125\ \mathrm{cm^3}$ . The larger is 15 cm tall. How tall is the smaller, and what is the ratio of their surface areas?
A model of a water tower is built to a scale of $1 : 20$ . The model's tank holds 5 litres. How many litres does the real tank hold?
A triangle of area $4\ \mathrm{cm^2}$ is enlarged with scale factor $- 3$ . Find the area of the image.

Show answersHide answers

$5 \times 9 = 45\ \mathrm{cm^2}$ .
$k = 2.5$ , $k^{2} = 6.25$ : $12 \times 6.25 = 75\ \mathrm{cm^2}$ .
$8 \times 25\,000^2 = 5 \times 10^9\ \mathrm{cm^2} = 500\,000\ \mathrm{m^2} = 50$ ha.
$k = 1.5$ : $0.8 \times 3.375 = 2.7$ litres.
$k^2 = \frac{36}{16}$ , so $k = \frac{6}{4} = 1.5$ ; a perimeter is a length: $20 \times 1.5 = 30$ cm.
$k = \sqrt[3]{\frac{125}{27}} = \frac{5}{3}$ : the smaller is $15 \times \frac{3}{5} = 9$ cm tall, and the surface areas are in the ratio $9 : 25$ .
$5 \times 20^3 = 5 \times 8000 = 40\,000$ litres.
$∣ k ∣^{2} = 9$ : $4 \times 9 = 36\ \mathrm{cm^2}$ .

1Twice as tall, eight times as much water

Twice as tall, eight times as much water