GCSEMathsOCRExam Questions9. Congruence & SimilarityArea & Volume of Similar Shapes

Area & Volume of Similar Shapes (OCR GCSE Maths)

Exam Questions

2 hours37 questions
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1
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A company makes monsters.

The company makes small monsters with a height of 20 cm.

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A small monster has a surface area of 300 cm2.

The company also makes large monsters with a height of 120 cm.

A small monster and a large monster are mathematically similar.

Work out the surface area of a large monster.

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2a
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The diagram shows two similar solids, A and B.

q2-easy-4-7-area-and-volume-of-similar-shapes-edexcel-gcse-maths

Solid A has a volume of 80 cm3.

Work out the volume of solid B.

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2b
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Solid B has a total surface area of 160 cm2.

Work out the total surface area of solid A.

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3
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In the diagram, AED and ABC are straight lines and BE is parallel to CD.

q13-paper6-nov2020-ocr-gcse-maths

The ratio of length AB to length BC is 2 : 3.
Triangle ABE has an area of 8cm2
.
Work out the area of triangle ACD.

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

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4
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A transport lorry consists of a cab and a trailer.
The trailer has a volume of 90m3.
Alfie makes a model of this lorry using a scale of 1 : 72.

Work out the volume of the trailer in Alfie’s model, giving your answer in cm3.

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

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5
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Toy building bricks are available in two sizes, small and large.

The small and large bricks are mathematically similar.

A small brick has volume 8cm3 and width 2.1cm.

A large brick has volume 15.625 cm3.

Calculate the width of a large brick.

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

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6
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Simon cuts the corners off a square piece of card to leave the regular octagon shown below.

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Simon makes a table top using the card as a model.
The sides of the table top are 8 times as long as the sides of the card model.

Find the ratio of the area of Simon’s table top to the area of the card model.

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7
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bold A andbold space bold B bold spaceare two similar solids.

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bold A has a volume of 1836 cm3
bold B has a volume of 4352 cm3

bold B has a total surface area of 1120 cm2
Work out the total surface area of bold A.

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

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8
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bold R bold spaceand bold S are two similar solid shapes.

Shape bold R has surface area 108 cm squared and volume 135 cm cubed
Shape bold S has surface area 300 cm squared

Work out the volume of shape bold S.

............................................... cm cubed

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9a
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bold A and bold B are two similar vases.

q17-paper1h-june2019-edexcel-igcse-maths

Vase bold Ahas height 24 cm.
Vase bold B has height 36 cm.

Vase bold A has a surface area of 960 cm squared

Work out the surface area of vase bold B.

........................cm squared

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9b
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Vase bold B has a volume of V space cm cubed

Find in terms of V , an expression for the volume, in space cm cubed, of vase bold A.

.........................cm cubed

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10
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Mathematically similar wooden blocks are made in a workshop.

There are small blocks and there are large blocks.

The volume of each small block is 300cm3

Given that the surface area of each small block : the surface area of each large block = 25 : 36

work out the volume of each large block.

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

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11
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The diagram shows two mathematically similar vases, bold A and bold B.

q16-paper2h-jan2019-edexcel-igcse-maths

bold A has a volume of 405 cm cubed
bold B has a volume of 960 cm cubed

bold B has a surface area of 928 cm squared

Work out the surface area of bold A.

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12
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The diagram shows two similar bottles, bold A and bold B.

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Bottle bold A  has surface area 240 space cm squared
Bottle bold B has surface area 540 space cm squared spaceand volume 2025 space cm cubed
Work out the volume of bottle bold A.

.......................cm cubed

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13
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A and B are similar solid cylinders.

base area of A : base area of B = 9 : 25


Complete these ratios.

curved surface area of A : curved surface area of B =................... : ................

height of A : height of B =.................... : ................

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14
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A and B are similar cuboids.

surface area of A : surface area of B = 16 : 25

Work out volume of A : volume of B

Circle your answer.

4 : 5 16 : 25 64 : 125 256 : 625

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15
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Solids X and Y are similar.

straight X has volume 64 cm3
straight Y has volume 343 cm3

The surface area of straight X is 176 cm2
Work out the surface area of straight Y.

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

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1
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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.

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2
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bold P and bold space bold Q are two triangular prisms that are mathematically similar.

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Prism bold P has triangle A B C as its cross section.
Prism bold Q has triangle D E F as its cross section.

A C = 6 cm
D F= 12 cm

The area of the cross section of prism bold P is 10 cm2.
The length of prism bold P is 15 cm.

Work out the volume of prism bold Q.

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3a
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A B C D and A E F G are mathematically similar trapeziums.
A E space equals space 5 space cm space E F space equals space 12 space cm space B C space equals space 18 space cm

Work out the length of  A B.

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3b
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Trapezium A E F G has an area of 36 cm2.

Work out the area of the shaded region.

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4
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Here are two similar solid shapes.

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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.

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5
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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.

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6
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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?

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7
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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

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8a
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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.

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8b
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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 colon f

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9
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The diagram consists of three mathematically similar shapes.
The heights of the shapes are in the ratio 1 : 4 : 5.

q17-paper5-nov2018-ocr-gcse-maths

Find the ratio

total shaded area : total unshaded area.

Give your answer in its simplest form.

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

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10
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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]

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11
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The three solids bold A bold comma bold space bold B and C are similar such that

the surface area of bold A space colon the surface area of bold B space equals space 4 space colon space 9

and

the volume of space bold B space colon spacethe volume of bold C space equals space 125 space colon space 343

Work out the ratio

the height of bold A space colon the height of bold C

Give your ratio in its simplest form.

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12
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A B C D E F and G H I J K L are regular hexagons each with centre O.

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G H I J K L is an enlargement of A B C D E F, with centre O and scale factor 2

The triangle O A B has an area of 5 cm2

Calculate the area of the shaded region.

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

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13
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A B C D E and A F G H J are regular pentagons.

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A E J space and space A B F spaceare straight lines.

E J space equals space 4 A E
The area of space A B C D E space is 8 space cm squared

Calculate the area of the shaded region.

.............................cm squared

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14a
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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

q23-paper-2h-nov-2019-aqa-gcse-maths

The total surface area of A is 62 cm2

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

Here is his working.

table row cell 10 space divided by space 5 space end cell equals cell space 2 space end cell row cell 62 space cross times space 2 space end cell equals cell space 124 space end cell row cell Total space surface space area space of space straight B space end cell equals cell space 124 space cm squared end cell end table

Make one criticism of Tim’s method.

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14b
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Volume of A space cross times space 125 over 8 equals Volume of C

Work out the value of x.

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15
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A pattern is made from two similar trapeziums.

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Show that the shaded area is 294 cm2

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1
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L M N is a right-angled triangle.

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Angle N L M= 90°
P Q is parallel to L M.

The area of triangle P N Q is 8 cm2
The area of triangle L P Q is 16 cm2

Work out the area of triangle L Q M.

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2
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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

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3
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The diagram shows two similar vases, bold A and bold B.

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The height of vase bold A is 9cm and the height of vase bold B is 13 cm.

Given that

surface area of vase bold A + surface area of vase bold B = 1800cm squared

calculate the surface area of vase bold A.

........................cm squared

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4
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A sphere of radius r has a surface area of 4 straight pi r squared and a volume of 4 over 3 straight pi r cubed.
A cylinder of radius r and height h has a curved surface area of 2 straight pi r h and a volume of straight pi r squared h.

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

volume space of space hemisphere space colon space volume space of space cylinder space equals space 1 space colon m

find the value of m.

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5
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A frustum is made by removing a small cone from a large cone.
The cones are mathematically similar.

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

fraction numerator volume space of space frustum over denominator volume space of space large space cone end fraction equals 98 over 125

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

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

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6
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bold A andbold space bold B spaceare two similar vases.

q17-january-2022-1h-edexcel-igcse-maths

Vase bold A has height 10 cm.
Vase bold B space has height 15 cm.

The difference between the volume of vase bold A and the volume of vase bold B is 1197 cm3

Calculate the volume of vase bold A



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

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7a
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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

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7b
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Calculate the area of the logo.
 

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

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7c
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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

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