Congruence & Similarity (Cambridge (CIE) IGCSE Maths: Extended): Exam Questions

A paperweight has height 4 cm and volume 38.4 cm³.
A mathematically similar paperweight has height 7 cm.  

Calculate the volume of this paperweight.  

 .......................................... cm³ 

Two mathematically similar containers have heights of 30 cm and 75 cm.
The larger container has a capacity of 5.5 litres.

Calculate the capacity of the smaller container.
Give your answer in millilitres.

............................................... ml

A model of a car has a scale 1:20.
The volume of the actual car is 12 m³.

Find the volume of the model.
Give your answer in cubic centimetres.  

......................................cm³

The diagram shows two glasses which are mathematically similar.
The larger glass has a capacity of 0.5 litres and the smaller glass has a capacity of 0.25 litres.
The height of the larger glass is 13 cm.

Calculate the height of the smaller glass.

........................................... cm

A, B, C and D are points on the circumference of the circle.
AC and BD intersect at X.

Complete the statement.

Triangle ADX is ..................................................... to triangle BCX. 

The area of triangle ADX is 36 cm² and the area of triangle BCX is 65.61 cm².
AX = 8.6 cm and DX = 7.2 cm. Find BX.

BX = .................................. cm

PQR is a triangle.
T is a point on PR and U is a point on PQ.
RQ is parallel to TU.

Explain why triangle PQR is similar to triangle PUT. Give a reason for each statement you make.

PT : TR = 4 : 3

i) Find the ratio PU : PQ.  

...................... : ...................... [1]

ii) The area of triangle PUT is 20cm². Find the area of the quadrilateral QRTU.   

.......................................... cm² [3]

Triangle ABC is mathematically similar to triangle PQR.
The area of triangle ABC is 16 cm².

Calculate the area of triangle PQR.

 .......................................... cm² 

The triangles are the cross-sections of prisms which are also mathematically similar.
The volume of the smaller prism is 320 cm³.  

Calculate the length of the larger prism.  

 ............................................ cm

Two cones are mathematically similar.
The total surface area of the smaller cone is 80 cm². The total surface area of the larger cone is 180 cm². The volume of the smaller cone is 168 cm³.

Calculate the volume of the larger cone.

............................................. cm³

The diagram shows two mathematically similar solid metal prisms.
The volume of the smaller prism is 648 cm³ and the volume of the larger prism is 2187 cm³.
The area of the cross-section of the smaller prism is 36 cm².

Calculate the area of the cross-section of the larger prism.  

............................................. cm² 

Two containers are mathematically similar.
The surface area of the larger container is 226 cm² and the surface area of the smaller container is 94 cm².
The volume of the larger container is 680 cm³.  

Find the volume of the smaller container.  

........................................... cm³ 

Two cups $A$ and $B$ are made from a thin plastic and shaped as cones.

• Cup $A$ has a radius of $3\mathrm{cm}$ and a height of $12\mathrm{cm}$.

• Cup $B$ is geometrically similar to cup $A$.

The capacity of cup $B$ is $252\mathrm{\pi }{\mathrm{cm}}^3$ more than cup $A$.

Find the radius of cup $B$.

A model of a statue has a height of 4cm.
The volume of the model is 12 cm³.
The volume of the statue is 40 500 cm³.

Calculate the height of the statue.

 ............................................ cm

$C$ lies on a circle with diameter $AD$.
$B$ lies on $AC$ and $E$ lies on $AD$ such that $BE$ is parallel to $CD$.
$AB$ = 21 cm, $CD$ = 18 cm and $BE$ = 13.5 cm.

Work out the radius of the circle.

............................................ cm

The diagram shows two shapes that are mathematically similar.
The smaller shape has area 52.5 cm² and the larger shape has area 134.4 cm².  

Calculate the value of $m$.  

$m=$................................................

The area of pentagon A is $73.5{\text{cm}}^2$.
The area of pentagon B is $6{\text{cm}}^2$.

Find the ratio    perimeter of pentagon A : perimeter of pentagon B   in its simplest form.  

......................... : .........................

Manjeet has two mathematically similar bottles in her bathroom.
The large bottle holds 1.35 litres and is 29.7 cm high. The small bottle holds 0.4 litres.

Calculate the height of the small bottle.

............................................ cm

The diagram shows two mathematically similar shapes with areas $295{\text{cm}}^2$ and $159.5{\text{cm}}^2$.
The width of the larger shape is $17\text{cm}$.

Calculate the width of the smaller shape.  

......................................... cm

The diagram shows a company logo made from a rectangle and a major sector of a circle.
The circle has centre O and radius OA.
OA = OD = 0.5 cm and AB = 1.5 cm.
E is a point on OC such that OE = 0.25 cm and angle OED = 90°.

Calculate the perimeter of the logo.  

............................................. cm

Calculate the area of the logo.  

............................................ cm²

A mathematically similar logo is drawn. The area of this logo is 77.44 cm².

Calculate the radius of the major sector in this logo.  

............................................. cm

A solid metal cone has radius 10 cm and height 36 cm.

Calculate the volume of this cone.

[The volume, $V$, of a cone with radius $r$ and height $h$ is  $V=\frac{1}{3}\mathrm{\pi }{r}^{2}h$.]  

......................................... cm³

The cone is cut, parallel to its base, to give a smaller cone.

The volume of the smaller cone is half the volume of the original cone.

The smaller cone is melted down to make two different spheres.

The ratio of the radii of these two spheres is 1 : 2.

Calculate the radius of the smaller sphere.

[The volume, $V$, of a sphere with radius $r$ is  $V=\frac{4}{3}\mathrm{\pi }{r}^3$.]

.......................................... cm