Basic Engineering Mathematics, Fifth Edition

Areas of common shapes 227

inthe middleof the garden. Find, correct tothe

nearest square metre, the area remaining.

2. Determine the area of circles having (a) a
   radius of 4cm (b) a diameter of 30mm (c) a
   circumference of 200mm.


3. An annulus has an outside diameter of 60mm
   and an inside diameter of 20mm. Determine
   its area.


4. If the area of a circle is 320mm^2 , find (a) its
   diameter and (b) its circumference.


5. Calculate the areas of the following sectors of
   circles.
   (a) radius 9cm, angle subtended at centre
   75 ◦.
   (b) diameter 35mm, angle subtended at
   centre 48◦ 37 ′.


6. Determine the shaded area of the template
   shown in Figure 25.23.

120mm

90mm

80mm

radius

Figure 25.23

7. An archway consists of a rectangular opening
   topped by a semi-circular arch, as shown in
   Figure 25.24. Determine the area of the open-
   ing if the width is 1m and the greatest height
   is 2m.

1m

2m

Figure 25.24

Here are some further worked problems of common
shapes.

Problem 17. Calculate the area of a regular

octagon if each side is 5cm and the width across the

flats is 12cm

An octagon is an 8-sided polygon. If radii are drawn

from the centre of the polygon to the vertices then 8

equal triangles are produced, as shown in Figure 25.25.

12cm

5m

Figure 25.25

Area of one triangle=

1

2

×base×height

=

1

2

× 5 ×

12

2

=15cm^2

Area of octagon= 8 × 15 =120cm^2

Problem 18. Determine the area of a regular

hexagon which has sides 8cm long

A hexagon is a 6-sided polygon which may be divided

into 6 equal triangles as shown in Figure 25.26. The

angle subtended at the centre of each triangle is 360◦÷

6 = 60 ◦. The other two angles in the triangle add up to

120 ◦and are equal to each other. Hence, each of the

triangles is equilateral with each angle 60◦and each

side 8cm.

4cm

8cm

8cm

608

h

Figure 25.26

Area of one triangle=

1

2

×base×height

=

1

2

× 8 ×h