GCSE Maths · Higher · Chapter 12: Similarity · Lesson 2

Area and volume scale factors

Grades 6–9About 50 minutesCalculator allowed42 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Two triangles are similar. A side of 6 cm matches 9 cm. Another side is 5 cm. Find its match.
[1 mark]
Answer
A2
Key Stage 3
Simplify $8 x + 2 y - 6 x - 4 y$ .
[1 mark]
Answer
A3
Key Stage 3
Expand $3 (3 x - 1)$ .
[1 mark]
Answer
A4
Key Stage 3
Write $21$ out of $50$ as a percentage.
[1 mark]
Answer

Grades 6–7

Your teacher works through each example with you. Then try the one beside it.

Example 1

These two rectangles are similar. Work out the area of the larger rectangle.

Not drawn accurately

Hint Find the length scale factor, then square it for area.

B1
These two triangles are similar. Their areas are $45$ cm² and $80$ cm². Work out the length $x$ .
[3 marks]
Not drawn accurately
Answer

Example 2

These two cylinders are similar. The smaller has a volume of $40$ cm³. Work out the volume of the larger.

Not drawn accurately

Hint Find the length scale factor, then cube it for volume.

B2
These two cones are similar. Their volumes are $192$ cm³ and $648$ cm³. Work out the height $h$ of the smaller cone.
[3 marks]
Not drawn accurately
Answer

Grades 6–8

C1
Two similar candles are made of the same wax. The smaller has a mass of $216$ g. Work out the mass of the larger, in grams.
[3 marks]
Not drawn accurately
Answer
C2
Two similar boxes are painted all over. It takes $0.4$ litres of paint to paint the smaller box. How much paint is needed for the larger?
[2 marks]
Not drawn accurately
Answer

C3
Two similar solids have volumes in the ratio $8 : 125$ . Write down the ratio of their surface areas.
[2 marks]
Answer
C4
$D E$ is parallel to $B C$ . The area of triangle $A B C$ is $75$ cm². Work out the area of triangle $A D E$ .
[3 marks]
Not drawn accurately
Answer
C5
A map has a scale of $1 : 25000$ . A lake has an area of $6$ cm² on the map. Work out the real area of the lake, in km².
[3 marks]
Answer
C6
Two similar solids have volumes $80$ cm³ and $270$ cm³. The larger has a surface area of $207$ cm². Work out the surface area of the smaller.
[3 marks]
Answer

Grades 7–9

D1
A model van is made to a scale of $1 : 20$ . The real van's load space is $1.6$ m³. Tom says, "The model's load space is $1.6 \div 20 = 0.08$ m³."
Explain why Tom is wrong, and work out the model's load space in cm³.
[4 marks]

D2
A cone-shaped glass, point down, is $18$ cm deep and holds $810$ cm³ when full. It has water $12$ cm deep in it. How much more water will fill it? You must show your working.
[4 marks]
Not drawn accurately
Answer
D3
These similar triangles have areas $18$ cm² and $50$ cm². Work out $x$ . You must show your working.
[4 marks]
Not drawn accurately
Answer

Grade 9 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

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