Ratio of Area and volume

Content

When two Triangles are Congruent?
Ratio of Area and Volume for similar figures
Examples

When two Triangles are Congruent?

All corresponding sides are equal. (S S S)

a = p, b = q, c = r

Two sides and in-between angle are equal. (S A S)

a = p, <B = <Q, c = r

Two angles and in-between side are equal. (A S A)

<A = <P, b = q, <C = <R

Both have right angles, equal hypotenuse, equal side. (R H S)

Ratio of Area and Volume for similar figures

Length = a : b = 2 : 3

Area = a² : b²

= 2² : 3² = 4 : 9

Volume = a³ : b³

= 2³ : 3³ = 8 : 27

Examples

Height of two similar cylinders are in the ratio 4 : 3.

Find ratios of their (a) surface areas (b) volumes.

Ratio of heights = 4 : 3

Ratio of surface areas = 4² : 3²= 16 : 9

Ratio of volumes = 4³ : 3³

= 64 : 27

Surface areas of two similar containers are in the ratio 9 : 25.

Find ratios of their (a) heights (b) volumes.

Ratio of surface areas = 9 : 25

Ratio of heights = Sqrt(9) : Sqrt(25) = 3 : 5

Ratio of volumes = 3³ : 5³ = 27 : 125

Question

A model of a cylindrical water tank is in the scale of 1: 30 (a) If the actual height of the water tank is 3 m, find the height of the model in cm. (b) if the base area of the model is 20 cm² , find the actual base area of the tank in m² (c) if the model has a volume of 2000 cm³ , find the actual volume of the tank in m³