Area & Volume of Similar Shapes (Edexcel IGCSE Maths A: Higher): Exam Questions

Fred is making two rectangular flower beds.

The dimensions of the larger rectangle will be three times the dimensions of the smaller rectangle.

There is going to be the same depth of soil in each flower bed. Fred needs 180 kg of soil for the smaller flower bed.

Work out how much soil Fred needs for the larger flower bed.

$\mathbf{P}$ and $\mathbf{Q}$ are two triangular prisms that are mathematically similar.

Prism $\mathbf{P}$ has triangle $ABC$ as its cross section.
Prism $\mathbf{Q}$ has triangle $DEF$ as its cross section.

$AC$ = 6 cm
$DF$= 12 cm

The area of the cross section of prism $\mathbf{P}$ is 10 cm².
The length of prism $\mathbf{P}$ is 15 cm.

Work out the volume of prism $\mathbf{Q}$.

$ABCD$ and $AEFG$ are mathematically similar trapeziums.$$\begin{gathered} AE=5\mathrm{cm} \\ EF=12\mathrm{cm} \\ BC=18\mathrm{cm} \end{gathered}$$

Work out the length of  $AB$.

Trapezium $AEFG$ has an area of 36 cm².

Work out the area of the shaded region.

Here are two similar solid shapes.

surface area of shape A : surface area of shape B = 3:4

The volume of shape B is 10 cm³

Work out the volume of shape A.
Give your answer correct to 3 significant figures.

Solid A and solid B are mathematically similar.
The ratio of the surface area of solid A to the surface area of solid B is 4: 9

The volume of solid B is 405 cm³.

Show that the volume of solid A is 120cm³.

Mark has made a clay model.
He will now make a clay statue that is mathematically similar to the clay model.

The model has a base area of 6cm²
The statue will have a base area of 253.5 cm²

Mark used 2kg of clay to make the model.

Clay is sold in 10kg bags.
Mark has to buy all the clay he needs to make the statue.

How many bags of clay will Mark need to buy?

Cone A and cone B are mathematically similar.
The ratio of the volume of cone A to the volume of cone B is 27:8

The surface area of cone A is 297 cm²

Show that the surface area of cone B is 132 cm²

The circumference of circle B is 90% of the circumference of circle A.

Find the ratio of the area of circle A to the area of circle B.

Square E has sides of length $e$ cm.
Square F has sides of length $f$ cm.

The area of square E is 44% greater than the area of square F.

Work out the ratio $e:f$

The three solids $\mathbf{A}\mathbf{,}\mathbf{B}$ and C are similar such that

the surface area of $\mathbf{A}:$ the surface area of $\mathbf{B}=4:9$

and

the volume of $\mathbf{B}:$the volume of $\mathbf{C}=125:343$

Work out the ratio

the height of $\mathbf{A}:$ the height of $\mathbf{C}$

Give your ratio in its simplest form.

$ABCDEF$ and $GHIJKL$ are regular hexagons each with centre $O$.

$GHIJKL$ is an enlargement of $ABCDEF$, with centre $O$ and scale factor 2

The triangle $OAB$ has an area of 5 cm²

Calculate the area of the shaded region.

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

$ABCDE$ and $AFGHJ$ are regular pentagons.

$AEJ$ and $ABF$are straight lines.

$EJ=4AE$
The area of $ABCDE$ is $8{\mathrm{cm}}^2$

Calculate the area of the shaded region.

.............................${\mathrm{cm}}^2$

Here are three similar cuboids, A, B and C.

A has length 5 cm, width 2 cm and height 3 cm
B has length 10 cm
C has length $x$ cm

The total surface area of A is 62 cm²

Tim wants to work out the total surface area of B.

Here is his working.

Make one criticism of Tim’s method.

Volume of $A\times \frac{125}{8}=$ Volume of C

Work out the value of $x$.

A pattern is made from two similar trapeziums.

Show that the shaded area is 294 cm²

The diagram consists of three mathematically similar shapes. The heights of the shapes are in the ratio 1 : 4 : 5.

Find the ratio

total shaded area : total unshaded area.

Give your answer in its simplest form.

total shaded area : total unshaded area ............ : ..........

Prism P and prism Q are similar. The ratio of the surface area of prism P to the surface area of prism Q is 1 : 3.

i) Jay says

The height of prism P is one third of the height of prism Q.

Explain why he is wrong.

[1]

ii) The volume of prism Q is 86 cm³.

Calculate the volume of prism P.

.................................................... cm³ [3]

$LMN$ is a right-angled triangle.

Angle $NLM$= 90°
$PQ$ is parallel to $LM$.

The area of triangle $PNQ$ is 8 cm²
The area of triangle $LPQ$ is 16 cm²

Work out the area of triangle $LQM$.

$\mathbf{A}$ and$\mathbf{B}$are two similar vases.

Vase $\mathbf{A}$ has height 10 cm.
Vase $\mathbf{B}$ has height 15 cm.

The difference between the volume of vase $\mathbf{A}$ and the volume of vase $\mathbf{B}$ is 1197 cm³

Calculate the volume of vase $\mathbf{A}$

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

The diagram shows two similar vases, $\mathbf{A}$ and $\mathbf{B}$.

The height of vase $\mathbf{A}$ is 9cm and the height of vase $\mathbf{B}$ is 13 cm.

Given that

surface area of vase $\mathbf{A}$ + surface area of vase $\mathbf{B}$ = 1800${\mathrm{cm}}^2$

calculate the surface area of vase $\mathbf{A}$.

........................${\mathrm{cm}}^2$

The total surface area of a solid hemisphere is equal to the curved surface area of a cylinder.

The radius of the hemisphere is $r$ cm.
The radius of the cylinder is twice the radius of the hemisphere.

Given that

$\mathrm{volume}\mathrm{of}\mathrm{hemisphere}:\mathrm{volume}\mathrm{of}\mathrm{cylinder}=1:m$

find the value of $m$.

A frustum is made by removing a small cone from a large cone. The cones are mathematically similar.

The large cone has base radius $r$ cm and height $h$ cm. Given that

$\frac{\mathrm{volume}\mathrm{of}\mathrm{frustum}}{\mathrm{volume}\mathrm{of}\mathrm{large}\mathrm{cone}}=\frac{98}{125}$

find an expression, in terms of $h$, for the height of the frustum.

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

A standard tin and a large tin are mathematically similar.
The volume of the large tin is 50% more than the volume of the standard tin.
Both tins are cylinders.
The radius of the standard tin is 10cm.

Calculate the radius of the large tin.

........................... cm 

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.  

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

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 

Solid A is similar to solid B

Here is some information about solid A and solid B

Work out the height of solid B
Give your answer as a decimal