Volumes
Mass, Volume and Density

Density is a term used to describe the mass of a unit of volume of a substance. For example, if the density of a metal is 2000 kg/m3, then 1 m3 of the substance has a mass of 2000 kg.

Mass, volume and density are related by the following equations.

Mass = Volume × Density

Volume =
and
Density =

The density of water is 1 gram/cm3 (or 1 g cm−3) or 1000 kg/m3 (or 1000 kg m−3).

Worked Examples

1

Find the mass of water in the fish tank shown in the diagram.

Show me

First calculate the volume of water.

V = 25 × 30 × 20
= 15000 cm3

Now use

Mass = Volume × Density
= 15000 × 1 (as density of water = 1g/cm3)
= 15000 grams
= 15 kg
2

The block of metal shown has a mass of 500 grams.

Find its density in

(a)

g/cm3,

Show me

First find the volume.

Volume = 5 × 8 × 10
= 400 cm3

Then use

Density =
Density =
= 1.25 g/cm3
(b)

kg/m3.

Show me

The process can then be repeated working in kg and m.

Volume = 0.05 × 0.08 × 0.1
= 0.0004 m3
Density =
Density =
= 1250 kg/m3

Exercises

A drinks carton is a cuboid with size as shown.

(a)

Find the volume of the carton.

cm³
(b)

If it contains 8 cm3 of air, find the volume of the drink.

cm³
(c)

Find the mass of the drink if it has a density of 1 gram/cm3.

g

A ream (500 sheets) of paper is shown in the diagram.

If the mass of the ream is 2.5 kg, find the density of the paper.

g/cm3 (to 3 d.p.)

A metal bar has a cross section with an area of 3 cm2 and a length of 40 cm. Its mass is 300 grams.

(a)

Find the volume of the bar.

cm3
(b)

Find the density of the bar.

g/cm3
(c)

Find the mass of another bar with the same cross section and length 50 cm.

g
(d)

Find the mass of a bar made from the same material, but with a cross section of area 5 cm2 and length 80 cm.

g

A bottle which holds 450 cm3 of water has a mass of 530 grams. What is the mass of the empty bottle?

g

The density of a metal is 3 grams/cm3. It is used to make a pipe with external radius of 1.5 cm and an internal radius of 0.5 cm.

(a)

Find the area of the cross section of the pipe.

cm2 (to 2 d.p.)
(b)

If the length of the pipe is 75 cm, find its mass.

g
(c)

Find the length, to the nearest cm, of a pipe that has a mass of 750 grams.

cm

The diagram shows the cross section of a metal beam. A 2 m length of the beam has a mass of 48 kg.

(a)

Find the density of the metal.

g/cm3

A second type of beam uses the same type of metal, but all the dimensions of the cross section are increased by 50%.

(b)

Find the length of a beam of this type that has the same mass as the first beam.

cm