Cambridge International Mathematics

376 Similarity (Chapter 18)

AREAS

The two circles shown are similar. Circle B is an enlargement of

circle A with scale factork.

Area of B=¼(kr)^2

=¼£k^2 r^2

=k^2 (¼r^2 )

=k^2 £area of A

We can perform a similar comparison for these similar rectangles.

Area of B=ka£kb

=k^2 ab

=k^2 £area of A

Using examples like this we can conclude that:

If a figure is enlarged with scale factorkto produce a similar figure then

the new area=k^2 £the old area.

Example 7 Self Tutor

Triangles ABC and PQR are similar

with AB=4cm and PQ=2cm.

The area of¢ABC is 20 cm^2.

What is the area of¢PQR?

Suppose we enlarge¢PQR to give¢ABC with scale factork.

k=

4

2

=2

) k^2 =4

So, area¢ABC=k^2 £area¢PQR

) 20 cm^2 =4£area¢PQR

) 5 cm^2 =area of¢PQR

Example 8 Self Tutor

Cylinders A and B have surface areas of

1600 cm^2 and 900 cm^2 respectively.

Given that the cylinders are similar, findx.

D AREA AND VOLUME OF SIMILAR SHAPES [4.5]

r

kr

AB

A B

a

b

ka

kb

BC

A

Q

P

4cm R

2cm

5cm xcm

A

B

IGCSE01

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y:\HAESE\IGCSE01\IG01_18\376IGCSE01_18.CDR Wednesday, 8 October 2008 10:15:02 AM PETER