Text
Unit C4 Section 3
Volumes of Pyramids, Cones and Spheres

The volumes of a pyramid, a cone and a sphere are found using the following formulae.

The proofs of these results are rather more complex and require mathematical analysis beyond the scope of this text.

Worked Examples

1

A cone and sphere have the same radius of 12 cm. Find the height of the cone if the cone and sphere have the same volume.

Suppose that the height of the cone is h cm.

Volume of cone = π × 122 × h = 48πh cm3
Volume of sphere = πr3 = π(12)3 = 2304π cm3

Since the volumes are equal

48πh = 2304π

Solving for h,

h = = = 48 cm

Exercises

Find the volume of the following containers:

(a)

a cylinder of base radius 3 cm and height 10 cm,

cm³ (to the nearest cm³)
(b)

a cone of base radius 5 cm and height 12 cm,

cm³ (to the nearest cm³)
(c)

a cone of base radius 7 cm and height 5 cm,

cm³ (to the nearest cm³)
(d)

a cone of base radius 1 m and height 1.5 m,

m³ (to 1 d.p.)
(e)

a cone of base radius 9 cm and slant height 15 cm,

cm³ (to the nearest cm³)
(f)

a cone of base diameter 20 cm and slant height 25 cm,

cm³ (to the nearest cm³)
(g)

a sphere of radius 6 cm,

cm³ (to the nearest cm³)
(h)

a hemisphere of radius 5 cm,

cm³ (to the nearest cm³)
(i)

a pyramid of square base 10 cm and height 15 cm,

cm³
(j)

a pyramid of rectangular base 8 cm by 6 cm and height 9 cm.

cm³

A wine glass is in the shape of a cone on a stem. The cylindrical tumbler is used to fill the wine glass. Find the diameter of the tumbler so that it has the same volume as the wine glass.

cm (to the nearest cm)

Find the volume of the following shapes made up of cones, hemispheres and cylinders.

Give your answers rounded to the nearest cm³

(a)
cm3
(b)
cm3
(c)
cm3
(d)
cm3

A cylinder is half filled with water as shown. A heavy sphere of diameter 2.5 cm is placed in the cylinder and sinks to the bottom.

By how much does the water rise in the cylinder?

cm (to 1 d.p.)

A marble paperweight consists of a cuboid and a hemisphere as shown in the diagram. The hemisphere has a radius of 4 cm.

Calculate the volume of the paperweight.

cm3 (to the nearest cm3)