GCSE · Congruence & similarity

Similar shapes & missing lengthsExam-style questions

Why equal angles force sides into a fixed ratio, finding the scale factor the right way round, similar triangles hidden inside one figure, and the area and volume scale factors k² and k³.

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25 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

1
Two solid statues are mathematically similar.

The smaller statue has volume $128$ cm³ and surface area $96$ cm².

The larger statue has volume $250$ cm³.

Work out the surface area of the larger statue.
Level 5 · 4 marksTry it
2
Two mathematically similar vases have surface areas $36$ cm² and $81$ cm².

The larger vase has volume $540$ cm³.

Work out the volume of the smaller vase.
Level 5 · 4 marksTry it
3
$A$ , $B$ and $C$ are three solid glass paperweights.

The paperweights are mathematically similar and are all made from the same type of glass.

The table gives information about the paperweights.
Paperweight $A$ Paperweight $B$ Paperweight $C$
Total surface area (cm²) $240$ $375$
Mass
(g) $810$ $240$
The height of paperweight $A$ is $9$ cm.

Work out the height of paperweight $C$ .

(4 marks)
Level 5 · 4 marksTry it
4
The diagram shows a solid cone.The cone is cut into three pieces, $P$ , $Q$ and $R$ , by two cuts parallel to its base.

$P$ is a cone. $Q$ and $R$ are frustums.

height of $P$ : height of $Q$ : height of $R$ $= 1 : 1 : 2$

Show that

volume of $P$ : volume of $Q$ : volume of $R$ $= 1 : 7 : 56$

(3 marks)
Level 5 · 3 marksTry it
5
$PQR$ is a triangle with a right angle at $Q$ . $S$ is the point on $PR$ such that $QS$ is perpendicular to $PR$ .

$PS : SR = 4 : 9$

Find the ratio $PQ : QR$

Give your answer in its simplest form.

(3 marks)
Level 5 · 3 marksTry it

	Paperweight $A$	Paperweight $B$	Paperweight $C$
Total surface area (cm²)	$240$	$375$
Mass (g)		$810$	$240$