Finding Missing Sides

Find the scale factor between similar shapes, then multiply to get a side on the big shape or divide to get a side on the small one.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024, IGCSE Additional Maths 0606 · 27 real past-paper questions below

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One Scale Factor for Every Side

In similar shapes, every side of the big shape is the matching small side times the same number: the scale factor.

Formula

scale factor = \frac{big side}{matching small side}

Find it from one pair of matching sides where you know both lengths. Then use it on the side you want.

Rule

Small to big: multiply by the scale factor. Big to small: divide by it.

Small to Big: Multiply

The most common 2-mark question: triangles $A B C$ and $D E F$ are similar. Find a side on the bigger one.

$A B = 4$ cm, $D E = 10$ cm, $A C = 3.2$ cm. Find $D F$ .

Worked example

1
$A B \leftrightarrow D E : 4 \to 10$ — $A B$ and $D E$ are matching sides and you know both lengths, so use them to find the scale factor.
2
$k = 10 \div 4 = 2.5$ — The scale factor is big over small, because the 4 has to grow into the 10.
3
$D F = A C \times 2.5$ — $D F$ is the side you want. It matches $A C$ and it is on the bigger triangle, so multiply by the scale factor.
4
$D F = 3.2 \times 2.5 = 8 cm$ — Work it out. 8 is bigger than 3.2, as it should be.

Common mistake

Finding the scale factor from two sides that do not match, like $\frac{10}{3.2}$ . Use the angle marks or the letter order to pair the sides.

Big to Small: Divide

The next 2-mark part asks for a side on the smaller shape. Same scale factor, but now you divide.

Same triangles: $E F = 12.5$ cm. Find $B C$ .

Worked example

1
$k = 10 \div 4 = 2.5$ — The triangles are the same, so the scale factor is still 2.5.
2
$B C = E F \div 2.5$ — $E F$ is on the bigger triangle and $B C$ is its match on the smaller one. Going big to small, divide by the scale factor.
3
$B C = 12.5 \div 2.5 = 5 cm$ — Work it out. 5 is smaller than 12.5, as it should be.

Common mistake

Multiplying anyway: $12.5 \times 2.5 = 31.25$ . A side of the small triangle cannot be longer than its partner in the big one.

Similar Quadrilaterals

The two-part question (2 marks each) uses two similar four-sided shapes, $A B C D$ and $E F G H$ . The letters are in matching order, so $A B$ matches $E F$ , $B C$ matches $F G$ , and so on.

$A B = 3$ , $B C = 4$ , $C D = y$ . $E F = x$ , $F G = 10$ , $G H = 20$ .

Worked example

1
$B C \leftrightarrow F G : 4 \to 10$ — The letters are in matching order, so $B C$ matches $F G$ . It is the only pair where both lengths are given.
2
$k = 10 \div 4 = 2.5$ — The scale factor is big over small.
3
$x = E F = A B \times 2.5 = 7.5$ — $E F$ matches $A B$ and is on the bigger shape, so multiply $A B = 3$ by the scale factor.
4
$y = C D = G H \div 2.5 = 8$ — $C D$ matches $G H$ and is on the smaller shape, so divide $G H = 20$ by the scale factor.

Key idea

Write the two shapes' sides in two rows, matched up. The pair with both numbers gives the scale factor.

Similar Solids: Any Matching Length

Solids work the same way. In similar solids every length has the same scale factor: height, radius, diameter and slant height.

Cylinder $A$ has height 8 cm and radius 3 cm. Cylinder $B$ is similar, with height 20 cm. Find the radius of $B$ .

Worked example

1
$h_{A} \leftrightarrow h_{B} : 8 \to 20$ — The heights are a matching pair and both are known.
2
$k = 20 \div 8 = 2.5$ — The scale factor is big over small.
3
$r_{B} = r_{A} \times 2.5$ — The radius is a length, so it uses the same scale factor. $B$ is the bigger cylinder, so multiply.
4
$r_{B} = 3 \times 2.5 = 7.5 cm$ — Work it out. 7.5 is bigger than 3, as it should be.

Common mistake

Dividing a height by a radius, like $\frac{20}{3}$ . Match like with like: height with height, radius with radius.

When the Scale Factor Is Untidy

Sometimes the scale factor does not come out neat. Do not round it. Keep it as a fraction, or keep every digit on your calculator.

Triangles $A B C$ and $D E F$ are similar. $A B = 2.1$ cm, $D E = 5.6$ cm and $A C = 4.5$ cm. Find $D F$ .

Worked example

1
$A B \leftrightarrow D E : 2.1 \to 5.6$ — A matching pair where you know both lengths.
2
$k = 5.6 \div 2.1 = 2.666 \dots$ — Big over small. The 6s never stop, so do not write 2.67.
3
$D F = 4.5 \times \frac{5.6}{2.1}$ — $D F$ matches $A C$ and is on the bigger triangle, so multiply by $k$ . Keep $k$ as the fraction $\frac{5.6}{2.1}$ , so nothing is lost.
4
$D F = 4.5 \times 5.6 \div 2.1$ — On the calculator, multiply by the top first, then divide by the bottom.
5
$D F = 12 cm$ — The calculator gives exactly 12. Using 2.67 would have given 12.015.

Common mistake

Rounding $k$ before you use it. Round only the final answer, and only if the question says how many decimal places or significant figures.

Key idea

Write the ratio you used, such as $\frac{5.6}{2.1}$ . If the final answer slips, the working can still earn marks.

Which Method? Read the Question

Find a matching pair with both lengths known. Then ask: is the side you want on the big shape or the small one?

You see	You do
Side on the bigger shape	multiply by the scale factor
Side on the smaller shape	divide by the scale factor
Letters in the same order	they tell you the matching sides
Cylinders or other solids	any matching lengths use the same scale factor
A scale factor that does not divide neatly	keep it as a fraction and round only at the end
A triangle inside a triangle	Similar Triangles
Prove they are similar	Proving Triangles Are Similar
An area, not a length	Area Factor of Similar Shapes

Key idea

Check the size: a side on the big shape must come out bigger than its partner.

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