Area & Volume ratios

Wednesday, April 19, 2023

a b

Ratio = a : b

a2b2

Area ratio = a2 : b2

a b

a3 b3

Volume ratio = a3 : b3

In General

Length scale factor = a : b

Area scale factor = a2 : b2

Volume scale factor = a3 : b3

Example

In the diagram drawn below triangle ABC is similar to triangle ADE and the area of ABC = 10 cm2

Find

(i) The length and area scale factor

(ii) The lettered lengths

(iii) The area ADE

(iv) The area BCDE

A C D

4 cm

B

E

12 cm

4 cm

6 cm y

x

Example

In triangle ABC, XY is parallel to BC and .

If the area of triangle AXY is 4 cm2, find the area of

triangle ABC.

23

AXAB

A

X

B C

Y

2

1

Example

The cube of side 1 cm is enlarged to give a cube of side 3 cm

a) State the length scale factor

b) The base area scale factor

c) State the scale factor of the volume.

Example • If the volume of the cone VMN is 4 cm3, find the volume of

a) The cone VPQ

b) The frustum MNQPP

N

Q

M

V

6cm 2cm

Example

Pamela has a fish-tank which is 30cm long and has a

capacity of 4.8 litres.

Rani has a similar fish-tank which is 45cm long.

What is the capacity of Rani’s fish-tank?

Example

Two similar cylindrical tins have base radii of 6 cm

and 8 cm respectively. If the capacity of the large tin

is 252 cm3, find the capacity of the small tin.

Example

Two similar spheres made of the same material have

weights of 32 kg and 108 kg respectively.

If the radius of the larger sphere is 9cm, find the

radius of the smaller sphere.

Example

Two solid spheres have surface areas of 5cm2 and 45

cm2 respectively and the mass of the smaller sphere is

1½ kg. Find the mass of the larger sphere.