Areas of common shapes 229
(a) its area in hectares (1 ha = 10 4 m 2 ).
(b) the length of fencing required, to the nearest 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.
3x
3x
x
x
(a)
(b)
Figure 25.29
Area of Figure 25.29(a) = (x)(x) = x
2
Area of Figure 25.29(b) = (3x)(3x) = 9x
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 proportional to the squares of corresponding linear
dimensions.
Problem 20. A rectangular garage is shown on a
building plan having dimensions 10 mm by 20 mm.
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 = 10 mm × 20 mm
= 200 mm
2
Since the areas of similar shapes are proportional to the
squares of corresponding dimensions,
True area of garage = 200 × (250)
2
= 12.5 × 10
6 mm
2
=
12.5 × 10 6
10 6
m
2
since 1 m 2 = 10 6 mm 2
= 12.5 m 2
Now try the following Practice Exercise
Practice Exercise 100 Areas of similar
shapes (answers on page 351)
1. The area of a park on a map is 500 mm 2 . If
the scale of the map is 1 to 40 000, determine the true area of the park in hectares
(1 hectare = 10 4 m 2 ).
2. A model of a boiler is made having an overall
height of 75 mm corresponding to an overall
height of the actual boiler of 6 m. If the area of
metal required for the model is 12 500 mm 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 diameter of 8 cm on the map. Calculate its area
in hectares, giving your answer correct to
3 significant figures. (1 hectare = 10 4 m 2 .)
(a) its area in hectares (1 ha = 10 4 m 2 ).
(b) the length of fencing required, to the nearest 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.
3x
3x
x
x
(a)
(b)
Figure 25.29
Area of Figure 25.29(a) = (x)(x) = x
2
Area of Figure 25.29(b) = (3x)(3x) = 9x
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 proportional to the squares of corresponding linear
dimensions.
Problem 20. A rectangular garage is shown on a
building plan having dimensions 10 mm by 20 mm.
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 = 10 mm × 20 mm
= 200 mm
2
Since the areas of similar shapes are proportional to the
squares of corresponding dimensions,
True area of garage = 200 × (250)
2
= 12.5 × 10
6 mm
2
=
12.5 × 10 6
10 6
m
2
since 1 m 2 = 10 6 mm 2
= 12.5 m 2
Now try the following Practice Exercise
Practice Exercise 100 Areas of similar
shapes (answers on page 351)
1. The area of a park on a map is 500 mm 2 . If
the scale of the map is 1 to 40 000, determine the true area of the park in hectares
(1 hectare = 10 4 m 2 ).
2. A model of a boiler is made having an overall
height of 75 mm corresponding to an overall
height of the actual boiler of 6 m. If the area of
metal required for the model is 12 500 mm 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 diameter of 8 cm on the map. Calculate its area
in hectares, giving your answer correct to
3 significant figures. (1 hectare = 10 4 m 2 .)