POP

E4.4 Similarity

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01 — Explanation

How do I know two shapes are similar, and how do I find a missing length?

Similar shapes have identical angles and all corresponding sides in the same ratio (the scale factor). To find a missing length, first identify two corresponding sides whose lengths you do know, find the scale factor between them, then apply it to the side you need:

\text{scale factor} = \frac{\text{corresponding length in shape B}}{\text{corresponding length in shape A}}

02 — Explanation

What's the relationship between length, area, and volume scale factors?

This is the key idea that separates Extended from Core similarity questions. If the length scale factor between two similar shapes is $k$ , then:

\text{area scale factor} = k^2 \qquad \qquad \text{volume scale factor} = k^3

This makes sense dimensionally: area is a product of two lengths (so the scale factor multiplies itself once more), and volume is a product of three lengths.

\frac{\text{Area of } A}{\text{Area of } B} = \left( \frac{\text{Length of } A}{\text{Length of } B} \right)^2 \qquad \frac{\text{Volume of } A}{\text{Volume of } B} = \left( \frac{\text{Length of } A}{\text{Length of } B} \right)^3

03 — Explanation

How do I go "backwards" — from an area or volume ratio to a length scale factor?

Reverse the power: square-root an area ratio to get the length scale factor, and cube-root a volume ratio.

k = \sqrt{\text{area ratio}} \qquad \qquad k = \sqrt[3]{\text{volume ratio}}

04 — Explanation

How do I explain why two triangles are similar, especially when they overlap or share a vertex?

State which angles are equal and why (e.g. "angle $A$ = angle $D$ (given)", "angle $B$ = angle $E$ (corresponding angles, $BC \parallel EF$ )"), then conclude "therefore triangle $ABC$ is similar to triangle $DEF$ (AA — two equal angles)." A common exam setup nests one triangle inside another, sharing a vertex angle — here, look for a line drawn parallel to one side of the larger triangle, since that parallel line automatically creates equal corresponding angles between the two triangles.

05 — Explanation

How do I find a missing area directly, without first working out the scale factor as a separate step?

You can set up the area ratio as an equation and solve for the unknown area in one go, using the length ratio squared as a multiplier: if shape $A$ has area $18\text{ cm}^2$ and shape $B$ 's corresponding length is $1.5$ times shape $A$ 's, then shape $B$ 's area is $18 \times 1.5^2 = 40.5\text{ cm}^2$ . This shortcut avoids writing out the scale factor formula explicitly each time, once you're confident with the method.

Exam tip

Never apply the length scale factor directly to an area or volume, and never apply an unsquared/uncubed ratio when the question asks for area or volume — this mix-up (using $k$ instead of $k^2$ or $k^3$ ) is the single most common error at Extended level on this topic.

06 — Worked example

Worked example

Two similar solids have lengths in the ratio $2:3$ . The smaller solid has a volume of $40\text{ cm}^3$ . Find the volume of the larger solid.

Length scale factor $k = \dfrac{3}{2}$ .
Volume scale factor $= k^3 = \left(\dfrac{3}{2}\right)^3 = \dfrac{27}{8}$ .
Volume of larger solid $= 40 \times \dfrac{27}{8} = 135 \text{ cm}^3$ .

07 — Worked example

Worked example: going backwards from an area ratio

Two similar triangles have areas of $12\text{ cm}^2$ and $75\text{ cm}^2$ . The smaller triangle has a base of $4\text{ cm}$ . Find the base of the larger triangle.

Area ratio $= \dfrac{75}{12} = 6.25$ .
Length scale factor $k = \sqrt{6.25} = 2.5$ .
Base of larger triangle $= 4 \times 2.5 = 10\text{ cm}$ .