Volume of spheres; density

Warm-up

Three quick questions from earlier lessons.

Discover

A ball of radius $r$ fits exactly in a cylinder of radius $r$ and height $2r$ . The ball fills $\tfrac{2}{3}$ of the cylinder. What is the ball's volume?

Not to scale

$\tfrac{4}{3}\pi r^3$
$4\pi r^2$

Key idea

Spheres, and how heavy a solid is.

Sphere:

V = \tfrac{4}{3}\pi r^3

(surface area

4\pi r^2

). Density = mass ÷ volume, so mass = density × volume.

Worked example

A steel ball of radius 3 cm, density 7.8 g/cm³.

V = 36\pi \approx 113.1

cm³

(4/3)π · 27.

m \approx 7.8 \times 113.1 \approx 882

g

Mass = density × volume.

Your turn, guided

Your turn: mass 200 g, volume 50 cm³. The density.

4

g/cm³

200 ÷ 50.

Practice ★

Find the volume of a sphere of radius 5 cm (exact, in terms of π).Not to scale
Find the volume of a sphere of radius 9 cm (exact, in terms of π).Not to scale

A block has mass 526 g and volume 27 cm³. Find its density (g/cm³).

Watch out!

Practice ★★

Find the volume of a hemisphere (half a ball) of radius 3 cm (exact, in terms of π).
Find the volume of a hemisphere (half a ball) of radius 5 cm (exact, in terms of π).

Exit ticket

Three questions to check you have got it.

Find the volume of a sphere of radius 5 cm (exact, in terms of π).Not to scale
Find the volume of a sphere of radius 9 cm (exact, in terms of π).Not to scale
Find the volume of a sphere of radius 8 cm (exact, in terms of π).Not to scale

Summary card

Spheres and density

$V = \tfrac{4}{3}\pi r^3$ ; density = mass/volume.

Watch out: $r^3$ for volume, $r^2$ for area.

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