Basic Engineering Mathematics, Fifth Edition

Areas of common shapes 229

(a) its area in hectares (1 ha= 104 m^2 ).

(b) the length of fencing required, to the near-

est metre, to completely enclose the plot

of land.

25.4 Areas of similar shapes

Figure 25.29 shows two squares, one of which has sides
three times as long as the other.

3 x

3 x

x

x

(a) (b)

Figure 25.29

Area of Figure 25.29(a)=(x)(x)=x^2

Area of Figure 25.29(b)=( 3 x)( 3 x)= 9 x^2

Hence, Figure 25.29(b) has an area (3)^2 ; i.e., 9 times the
area of Figure 25.29(a).
In summary,the areas of similar shapes are pro-
portional to the squares of corresponding linear
dimensions.

Problem 20. A rectangular garage is shown on a

building plan having dimensions 10mm by 20mm.

If the plan is drawn to a scale of 1 to 250, determine

the true area of the garage in square metres

Area of garage on the plan=10mm×20mm

=200mm^2

Since the areas of similar shapes are proportional to the

squares of corresponding dimensions,

True area of garage= 200 ×( 250 )^2

= 12. 5 × 106 mm^2

=

12. 5 × 106

106

m^2

since 1m^2 = 106 mm^2

=12.5m^2

Now try the following Practice Exercise

PracticeExercise 100 Areas of similar

shapes (answers on page 351)

1. The area of a park on a map is 500mm^2 .If
   the scale of the map is 1 to 40000, deter-
   mine the true area of the park in hectares
   (1hectare= 104 m^2 ).


2. A model of a boiler is made having an overall
   height of 75mm corresponding to an overall
   height of the actual boiler of 6m. If the area of
   metal required for the model is 12500mm^2 ,
   determine, in square metres, the area of metal
   required for the actual boiler.


3. The scale of an Ordnance Survey map is
   1:2500. A circular sports field has a diam-
   eter of 8cm on the map. Calculate its area
   in hectares, giving your answer correct to
   3 significant figures. (1 hectare= 104 m^2 .)