Cambridge International Mathematics

Similarity (Chapter 18) 377

Surface area of B=k^2 £surface area of A

) 900 =k^2 £ 1600

) 169 =k^2

) k=^34 fk> 0 g

Now the radius of B=k£the radius of A

) x=^34 £ 5

) x=3: 75

VOLUMES

Volume of box B

=ka£kb£kc

=k^3 abc

=k^3 £volume of box A

In general:

If the lengths of a solid are enlarged with scale factorkto produce a similar figure then

the new volume=k^3 £the old volume.

Example 9 Self Tutor

These two cylinders are similar with heights

2 cm and 4 cm respectively.

Cylinder A has volume 10 cm^3.

Find the volume of cylinder B.

Suppose we enlarge cylinder A to give cylinder B.

) k=^42 =2

) volume of B=k^3 £volume of A

=8£ 10 cm^3

=80cm^3

Example 10 Self Tutor

A and B are similar cylinders with areas of

ends 9 cm^2 and 25 cm^2.

Find the ratio of their volumes.

A

B

a b

c

ka

kb

kc

B 4cm

A 2cm

25 cmX

B

9cmX

A

Consider the reduction of cylinder A to give cylinder B.

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