Area Factor of Similar Shapes

Use the square of the length scale factor to find areas of similar shapes, and go back from an area ratio to a length.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1 · 35 real past-paper questions below

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Why Area Uses k Squared

Two shapes are similar when every length is multiplied by the same scale factor $k$ .

Double the sides of a square and four copies of the old square fit inside. Length and width both double, so the area is $2 \times 2 = 4$ times as big.

Length factor	Area factor
$2$	$4$
$3$	$9$
$1.5$	$2.25$

Rule

Area factor = $k^{2}$ . This works for any area: faces, surface areas, even the area of a label.

Key idea

Pizza deal: a 40 cm pizza is twice as wide as a 20 cm pizza, but it gives you four times as much to eat. The length factor is $2$ , so the area factor is $2^{2} = 4$ . If it costs less than four small ones, the big one wins.

Area From Two Lengths

The 2-mark question gives two matching lengths and one area. Two similar vases are 8 cm and 12 cm tall. The small one has surface area 320 cm².

Worked example

1
$k = \frac{12}{8} = 1.5$ — The heights are a matching pair. Big over small gives the length factor.
2
$k^{2} = 1.5^{2} = 2.25$ — Area grows in two directions, length and width, so the area factor is the length factor squared.
3
$320 \times 2.25 = 720 cm^{2}$ — Multiply the small vase's area by the area factor. The big vase has the bigger area, as it should.

Common mistake

Multiplying 320 by 1.5 to get 480. Area needs the factor squared.

Length From Two Areas

The 2-mark question can go backwards: two areas are given and you want a length. Undo the square with a square root.

Similar shapes have areas 36 cm² and 225 cm². A side of the small one is 4 cm.

Worked example

1
$\frac{225}{36} = 6.25$ — Big area over small area gives the area factor.
2
$k = 6.25 = 2.5$ — The area factor is $k^{2}$ , so the square root undoes it and gives the length factor.
3
$side = 4 \times 2.5 = 10 cm$ — A length uses $k$ , not the area factor. The big shape is bigger, so multiply.

Common mistake

Multiplying the side by 6.25. That is the area factor. Lengths use $k$ , so take the square root first.

Sums and Shaded Parts

The 4-mark question gives the total of two areas. Call the small area $A$ .

Vases 4 cm and 6 cm tall. Together their surface areas make 1300 cm².

Worked example

1
$k = \frac{6}{4} = 1.5$ — The heights are a matching pair. Big over small gives the length factor.
2
$k^{2} = 1. 5^{2} = 2.25$ — Square it to get the area factor.
3
$A + 2.25 A = 1300$ — The big area is 2.25 lots of the small area $A$ , so the two areas together are $A$ plus 2.25 $A$ .
4
$3.25 A = 1300$ — Add the lots: 1 lot plus 2.25 lots is 3.25 lots.
5
$A = 1300 \div 3.25 = 400 cm^{2}$ — Divide by 3.25 to find one lot. That is the small vase.
6
$400 + 2.25 \times 400 = 400 + 900 = 1300$ — Check: the big vase is 900 cm², and the two add back to 1300.

Common mistake

Answering 1300 ÷ 2.25. The total is 3.25 lots of the small area, not 2.25.

Shaded Part and Area Ratios

A shaded part between a big shape and a similar small shape inside it is often asked as a ratio. Turn the length ratio into an area ratio first.

The heights of the two shapes are 2 cm and 3 cm. Find shaded area : small area.

Worked example

1
$lengths 2 : 3$ — The heights are a matching pair, small first.
2
$areas 2^{2} : 3^{2} = 4 : 9$ — Area needs the factor squared, so square each number in the length ratio.
3
$shaded = 9 - 4 = 5$ — The shaded part is the big shape take away the small shape, so subtract the parts.
4
$shaded : small = 5 : 4$ — Write the ratio in the order asked: shaded first, then small.

Key idea

Add a real area to check. If the big shape is 90 cm², one part is $90 \div 9 = 10$ cm². The small shape is 40 cm² and the shaded part is 50 cm², and $90 - 40 = 50$ .

Common mistake

Using $3 - 2 = 1$ for the shaded part. Square the lengths first, then subtract the areas.

Map and Plan Scales for Area

Map and plan questions give a scale such as $1 : 50000$ and ask for an area. The scale is a length factor, so the area uses it squared.

On a map with scale $1 : 50000$ , a lake has area 8 cm². Find the real area in km².

Worked example

1
$1 cm \to 50000 cm$ — The scale says every 1 cm on the map is 50 000 cm on the ground.
2
$50000 cm = 0.5 km$ — 100 000 cm make 1 km, so 50 000 cm is half a km. Change the length to km first, so the answer comes out in km².
3
$area factor = 0.5^{2} = 0.25$ — 1 cm² on the map is 0.5 km by 0.5 km, so square the length.
4
$8 \times 0.25 = 2 km^{2}$ — Multiply the map area by the area factor.

Going the other way, the plan is smaller. A hall floor is 600 m² and the plan scale is $1 : 40$ . Find the area on the plan in cm².

Worked example

1
$area factor = (\frac{1}{40})^{2} = \frac{1}{1600}$ — The plan is smaller, so each length is divided by 40. Square it for area: divide by 1600.
2
$600 m^{2} = 600 \times 10000 = 6000000 cm^{2}$ — 1 m² is 100 cm by 100 cm, which is 10 000 cm². Change to cm² so the answer comes out in cm².
3
$6000000 \div 1600 = 3750 cm^{2}$ — Divide by the area factor.

Common mistake

Multiplying the area by 50 000 or 40. That is the length factor. For area, square it, or convert the length first and then square.

Percentage Change in Area

The 2-mark question: every length goes up by 20%. What happens to the area?

Worked example

1
$k = 1 + \frac{20}{100} = 1.2$ — A 20% increase gives 120% of the old length, which is a multiplier of 1.2.
2
$k^{2} = 1. 2^{2} = 1.44$ — Square it to get the area factor.
3
$1.44 = 144%$ — As a percentage, 1.44 is 144% of the old area.
4
$144% - 100% = 44% increase$ — Take away the original 100% to see the increase.

It works downwards too. Every length is cut by 30%. Find the percentage change in area.

Worked example

1
$k = 1 - \frac{30}{100} = 0.7$ — A 30% decrease leaves 70% of the old length, which is a multiplier of 0.7.
2
$k^{2} = 0. 7^{2} = 0.49$ — Square it to get the area factor.
3
$0.49 = 49%$ — The new area is 49% of the old area.
4
$100% - 49% = 51% decrease$ — The area has lost 51%, not 30%.

Common mistake

Saying the area goes up by 20% too, or by 40%. Square the factor, then read off the percentage.

Which Method? Read the Question

Look at what you are given: lengths or areas.

You see	You do
Two lengths, find an area	square the ratio
Two areas, find a length	square root the ratio
A total of two areas	call the small one A
A shaded ring or strip	big area minus small area
A ratio of shaded to unshaded	square the length ratio, then subtract the areas
A map or plan scale, find an area	change the lengths first, then square the factor
Lengths up by a percentage	square the factor
A volume or capacity	Volume Factor of Similar Shapes
Only a missing side	Finding Missing Sides

Key idea

Area in m² but lengths in cm? Work in cm², then see Area and Volume Conversions.

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