Similarity Worked ExamplesIGCSE Maths 0580

On a June 2025 question finding a side of similar triangles, the examiner report noted that those who went wrong "did not demonstrate an understanding of how to apply proportion": the same scale factor must multiply every side. Pair corresponding sides — the one opposite the same angle in each triangle. Matching sides by size alone fails whenever the triangles are drawn in different orientations.

Two pairs of equal angles proves similarity — you never need all three. Name the reason for each equal pair; 'they look the same' earns nothing. Equal angles give similarity, equal sides would give congruence.

Multiplying the area by $1.5$ instead of $2.25$ gives $30$ and is the standard error. If lengths are $1.5$ times bigger, the shape is $1.5$ times wider and $1.5$ times taller — hence $1.5^2$.

Keep the factor as the fraction $\frac{27}{8}$ — cancelling $96 \div 8 = 12$ first keeps the arithmetic exact and avoids decimals. Remember the three factors: lengths $k$, areas $k^2$, volumes $k^3$. Then sense-check the direction: on a June 2025 question about the volumes of similar prisms, the examiner report noted candidates who "did not appear to have considered whether the result was required to be bigger or smaller than the original value." Here the larger cone must hold more than $96\text{ cm}^3$.

Always return to the linear ratio before moving between area and volume. Going straight from $9:25$ to a volume ratio by cubing gives $729 : 15625$, which is wrong — that would be $k^6$.

Part (b) needs the factor squared and a squared unit conversion — both are easy to forget. On a June 2025 question about the area on a scale plan, very few candidates were correct: the examiner report found that candidates were unsure whether to multiply or divide by the scale, and even fewer squared it. Sense-check: an area under a square kilometre is plausible for a forest $2\text{ km}$ across a map; a value in the billions would mean the units were never converted.

The most common error is to treat $DB$ as the side matching $AD$ and write $DB = 6 \times \frac{5}{3} = 10$. The two triangles are $ADE$ and $ABC$, so $AD$ matches the whole of $AB$. Drawing the two triangles separately, side by side, makes the matching sides easy to see.

When the smaller triangle is "turned round" inside the larger one, matching sides by their position on the page goes wrong. A common error is $\frac{AD}{6} = \frac{9}{6}$, giving $AD = 13.5\text{ cm}$ — longer than $AC$ itself, which should ring an alarm. Listing the equal angles and writing the vertices in matching order fixes this. In part (c), $\frac{2}{3}$ of the area (a length factor) is the other usual slip.

The usual error in part (c) is to halve the height ($6\text{ cm}$): a cone only half as tall holds just $\frac{1}{8}$ of the volume. Going from a volume factor back to a length factor needs a cube root, and from a length factor to an area factor needs squaring. Rounding $k$ to $0.79$ too early moves the final answers in part (d) away from the accepted range — the March 2026 examiner report reminded candidates that they "must remember to use more than 3 figures in their intermediate calculations".

Capacities are volumes, so the length factor is a cube root. On a similar question, the March 2026 examiner report found that "Most candidates used the volume scale factor correctly to find the height of the smaller bottle." The usual slips are forgetting to change litres to millilitres and dividing the capacities straight into the height. Check the direction: the larger jar must be taller than $12\text{ cm}$.

A common error is to scale the height by the area ratio itself, giving $5 \times \frac{243}{75} = 16.2\text{ cm}$ — that treats an area factor as if it were a length factor. Dividing $243$ and $75$ by $3$ gives $81$ and $25$, both perfect squares, which is the signal that the numbers were chosen for a non-calculator square root.

The key idea is to complete the pyramid. On a similar question, the June 2025 examiner report found that "Few recognised that this question could be answered using the volumes of similar solids", and some assumed a length of the removed pyramid instead of using the scale factor. A related June 2025 report on a cone frustum noted that "Some mistakenly used the height of the frustum instead of the height of the small cone" — here $10\text{ cm}$ is the frustum, not either pyramid.