Volume of pyramids and cones

Warm-up

Three quick questions from earlier lessons.

Discover

A cone and a cylinder have the same base and height. How many cones of water fill the cylinder?

Not to scale

3
2

Not to scale

Tilt them to compare. How many full cones of water fill the cylinder?

3
2

Key idea

Pointed solids hold one third.

For a pyramid or cone:

V = \tfrac{1}{3}Bh

. Cone:

V = \tfrac{1}{3}\pi r^2 h

.

Worked example

A square pyramid, base 6 m, height 10 m.

B = 36

Base area.

V = \tfrac{1}{3} \cdot 36 \cdot 10 = 120

m³

One third of Bh.

Your turn, guided

Your turn: a cone with $r = 3$ , $h = 4$ (in terms of π).

12\pi

⅓ · 9π · 4.

Practice ★

Find the volume of a cone with radius 6 cm and height 6 cm (exact, in terms of π).Not to scale
Find the volume of a cone with radius 6 cm and height 12 cm (exact, in terms of π).Not to scale

A pyramid has a square base of side 4 m and height 15 m. Find its volume (m³).Not to scale

Watch out!

Practice ★★

A cone and a cylinder have the same radius 5 and the same height. What fraction of the cylinder's volume is the cone's?

Boss problem. An ice cream cone has radius 3 cm and height 8 cm, topped with a hemisphere of ice cream of the same radius. Give exact answers in terms of π.

Exit ticket

Three questions to check you have got it.

Find the volume of a cone with radius 6 cm and height 6 cm (exact, in terms of π).Not to scale
Find the volume of a cone with radius 6 cm and height 12 cm (exact, in terms of π).Not to scale
Find the volume of a cone with radius 4 cm and height 9 cm (exact, in terms of π).Not to scale

Summary card

Pyramids and cones

$V = \tfrac{1}{3}Bh$ .

Watch out: do not forget the ⅓.

About this lesson