OCR GCSE Maths
Topic questions Ratio, proportion and rates of change Similarity, lengths, areas and volumes
31 questions · 126 marks · about 2 hours
Also involves Powers and roots
The diagram shows two similar cuboids, X and Y.
Cuboid X has a square base with area 16 cm2 and height 5 cm. Cuboid Y has a square base with area 36 cm2.
Work out the height of cuboid Y. You must show your working.
Also involves Forming and solving equations
The diagram shows two similar triangles, AEB and ADC.
AB = 8 cm, BE = 5 cm and CD = 17.5 cm
Work out the length of BC.
Triangles ABC and PQR are mathematically similar. Angle ACB = Angle PRQ. Angle ABC = Angle PQR.
The perimeter of triangle PQR is 99 cm.
Find the length of PQ.
Also involves Forming and solving equations · Ratio
Triangles ABC and PQR are mathematically similar. Angle ACB = Angle PRQ. Angle ABC = Angle PQR.
The perimeter of triangle PQR is 99 cm.
Find the length of PQ.
Two ornaments, A and B, are mathematically similar. The table shows information about the two ornaments.
| Ornament A | Ornament B | |
| Height (m) | h | 12 |
| Surface area (m2) | 216 | A |
| Volume (m3) | 240 | 3750 |
Find the value of h and the value of A. You must show your working.
The diagram shows two straight lines AB and DC that intersect at P. DA is parallel to BC.
Complete these statements to show that triangle PAD is similar to triangle PBC.
ADP BCP alternate Angle........ = angle........ because they are........ angles
APD Angle........ = angle........ because they are........ angles
Angle........ = angle........ because they are........ angles
Triangle PAD is similar to triangle PBC because........
Also involves Indices
Two prisms, A and B, are mathematically similar.
The ratio of the volume of prism A to the volume of prism B is 8: 27. The height of prism A is 6 cm.
Work out the height of prism B.
Also involves Scale factors, scale diagrams and maps
The measurements, in centimetres, of two triangles are shown in the table.
| Side 1 | Side 2 | Side 3 | |
| Triangle 1 | 2.5 | 3.2 | 4 |
| Triangle 2 | 7 | 8.96 | 11.2 |
Are the two triangles mathematically similar? Show how you decide.
Are these two triangles definitely congruent? Give a reason.
Two bottles are mathematically similar.
The small bottle holds 0.5 litres and has a height of 35 cm. The large bottle holds 2 litres.
Calculate the height of the large bottle.
On the grid are seven triangles, labelled A to G.
Complete each statement by writing the letter of the correct triangle.
Triangle A is congruent to triangle........
Triangle B is mathematically similar to triangle........
Triangles ABC and DEF are mathematically similar. Angle ABC = Angle DEF.
Calculate the perimeter of triangle ABC.
Points A, B, C and D lie on the circumference of a circle. Line AC intersects line BD at point E.
Prove that triangle AED is similar to triangle BEC.
Triangles ABC and DEF are mathematically similar. Angle ABC = Angle DEF.
Calculate the perimeter of triangle ABC.
Two pyramids, A and B, are mathematically similar.
Pyramid A has surface area 12 cm2 and volume 8 cm3. Pyramid B has surface area 75 cm2.
Work out the volume of pyramid B. You must show your working.
These two triangles are mathematically similar.
Work out the value of x.
The diagram shows triangles ABC and ADE.
B lies on AD and C lies on AE. BC is parallel to DE.
Complete these statements to show that triangles ABC and ADE are similar.
Angle ABC = angle ADE because they are corresponding angles.
Angle ACB = angle........ because........
Triangles ABC and ADE are similar because........
Two solid ornaments are mathematically similar. The larger ornament is twice as tall as the smaller ornament.
The smaller ornament has a volume of 50 cm3.
Work out the volume of the larger ornament.
Also involves Ratio
In the diagram, AED and ABC are straight lines and BE is parallel to CD.
The ratio of length AB to length BC is 2: 3. Triangle ABE has an area of 8 cm2.
Work out the area of triangle ACD.
Also involves Converting between units
A transport lorry consists of a cab and a trailer. The trailer has a volume of 90 m3. 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.
Alfie paints his model lorry. He has eight colours available.
He decides to paint the cab in one colour and the trailer in a different colour.
He then wants to paint his name on the trailer. The name must be in a different colour to the trailer.
In how many different ways can Alfie paint his model lorry?
Also involves Ratio
Are these two triangles mathematically similar? Show how you decide.
In the diagram AB is parallel to CD. AED and BEC are straight lines.
Prove that triangle ABE is similar to triangle CDE.
Also involves Powers and roots
Toy building bricks are available in two sizes, small and large. The small and large bricks are mathematically similar.
A small brick has volume 8 cm3and width 2.1 cm. A large brick has volume 15.625 cm3.
Calculate the width of a large brick.
Also involves Powers and roots
Two cylinders, A and B, are mathematically similar.
Cylinder A has volume 2400 cm3 and height 12 cm. Cylinder B has volume 750 cm3.
Find the height of cylinder B. Give your answer correct to an appropriate degree of accuracy.
Also involves Ratio
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........:........
Here are three similar triangles.
Work out the value of x.
The diagram shows two right-angled triangles, OAB and OCD.
Work out the length of BD.
The diagram below shows two right-angled triangles.
Prove that triangles PQS and QRS are similar.
Percy sells paint in standard tins and large tins. The standard tin covers 40 m2 and the large tin covers 60 m2.
Percy publishes this chart showing the area that can be covered with each tin of paint.
Explain why the chart is misleading.
The standard tin and the 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 10 cm.
Calculate the radius of the large tin.
Also involves Properties of triangles and quadrilaterals
ABCD and PQRS are rectangles. O is the centre of both rectangles.
AC is a straight line passing through P, O and R. BD is a straight line passing through Q, O and S.
PQ = 6 cm and QR = 2 cm. The perimeter of rectangle ABCD is 40 cm.
Work out the length and width of rectangle ABCD.
Also involves Converting between units
A model railway is built using the scale 1: 87.
On the model railway, the distance between the rails is 16.5 mm.
Calculate, in metres, the distance between the rails for a full-size train.
The volume of a full-size train carriage is 220 m³. Trevor calculates the volume of a model train carriage to be 334 cm³ correct to 3 significant figures.
Is Trevor’s calculation correct? Show how you decide.
The diagram shows triangle ABC with D on AC and E on AB. DE is a straight line.
AD = 28 m, AE = 41 m, DE = 22 m and BC = 64 m.
Calculate the length CD.
Triangle A is drawn on the grid.
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.
Jay says
The height of prism P is one third of the height of prism Q.
Explain why he is wrong.
The volume of prism Q is 86 cm3.
Calculate the volume of prism P.
Memes
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