Basic Engineering Mathematics, Fifth Edition

246 Basic Engineering Mathematics

Alternatively, fromthe question,r=30mm=3cmand

h=80mm=8cm. Hence,

volume=

1

3

πr^2 h=

1

3

×π× 32 × 8 =75.40cm^3

Problem 13. Determine the volume and total

surface area of a cone of radius 5cm and

perpendicular height 12cm

The cone is shown in Figure 27.12.

h

12cm

r5cm

l

Figure 27.12

Volume of cone=

1

3

πr^2 h=

1

3

×π× 52 × 12

=314.2cm^3

Total surfacearea=curved surfacearea+areaof base

=πrl+πr^2

From Figure 27.12, slant heightlmay be calculated

using Pythagoras’ theorem:

l=

√

122 + 52 =13cm

Hence,total surface area=(π× 5 × 13 )+(π× 52 )

=282.7cm^2.

27.2.6 Spheres

For the sphere shown in Figure 27.13:

Volume=

4

3

πr^3 and surface area= 4 πr^2

r

Figure 27.13

Problem 14. Find the volume and surface area of

a sphere of diameter 10cm

Since diameter=10cm, radius,r=5cm.

Volume of sphere=

4

3

πr^3 =

4

3

×π× 53

=523.6cm^3

Surface area of sphere= 4 πr^2 = 4 ×π× 52

=314.2cm^2

Problem 15. The surface area of a sphere is

201 .1cm^2. Find the diameter of the sphere and

hence its volume

Surface area of sphere= 4 πr^2.

Hence, 201.1cm^2 = 4 ×π×r^2 ,

from which r^2 =^201.^1

4 ×π

= 16. 0

and radius,r=

√

16. 0 = 4 .0cm

from which,diameter= 2 ×r= 2 × 4. 0 =8.0cm

Volume of sphere=

4

3

πr^3 =

4

3

×π×( 4. 0 )^3

=268.1cm^3

Now try the following Practice Exercise

PracticeExercise 106 Volumes and surface

areas of commonshapes (answerson

page 351)

1. If a cone has a diameter of 80mm and a
   perpendicular height of 120mm, calculate
   its volume in cm^3 and its curved surface
   area.


2. A square pyramid has a perpendicular height
   of 4cm. If a side of the base is 2.4cm long,
   find the volume and total surface area of the
   pyramid.