

Measuring volume and capacity  Volume
and capacity measurement facts 
 Summary of volume formulae of some shapes
 Quick quiz 
Measuring volume
and capacity
Volume
is the amount of space an object occupies whereas capacity
is the amount of a fluid that an object can hold, i.e., capacity measures
the "internal volume" of a container. A jug has both volume
and capacity since it takes up space and can also hold an amount of
liquid. A block of wood has volume but no capacity because it takes
up space but cannot hold anything. These terms are often confused.
There
are two sets of (connected) units. Litres are measured for volume
of liquids and for the capacity of containers. Cubic metres, cubic
centimetres etc are used for volumes of objects, spaces, etc.
Volume
and capacity measurement facts
VOLUME 
SI
unit: cubic metre (m^{3}) 
1000
cubic centimetres (cm^{3}) 
=
1 cubic decimetre (dm^{3}) 

=
1 litre (L)

1
cm^{3 }= 1 L


1000
cubic decimetres (dm^{3}) 
=
1 cubic metre (m^{3})


=
1 kilolitre (kL) 
Alternatively,
especially for gases and liquids:
1
cubic centimetre (cm^{3}) 
=
1 millilitre (mL) 
1000
millilitres (mL) 
=
1 litre (L) 
1000
litres (L) 
=
1 kilolitre (kL) 

=
1 cubic metre (m^{3}) 
1000
kilolitres (kL) 
=
1 megalitre (ML) 
Summary
of volume formulae of some shapes:
Volume
of a cube = L
x
L
x
L
= L^{3}

To
find the volume of a cube we need to multiply together its 3 dimensions
of length, width and height. As they are the same for a cube, we can
simplify the formula to, the volume of a cube is its length cubed.
Finding
the volume of a prism
What
is a prism?
To
understand how to find the volume of a prism we first need to understand
what a prism is!
These
solids are prisms


Prisms
are special solids. They have two polygonal bases joined
by parallelograms. The polygonal bases give the uniform
crosssections. This means that if we cut a prism through
parallel to the base, each cut piece will be exactly the
same size and shape. Can you see why all these solids are
prisms? 

Example
1: Is this solid a prism?



Figure
1

Figure
2

Figure
3

Thinking

To
find out if this solid is a prism, we need to imagine cutting
the solid through parallel to a face to see if the faces of
the cut will be exactly the same size and shape if I take several
cuts parallel to that face.
Looking
at figures 1 and 2, we can see that if the solid is cut several
times parallel to the face it is sitting on, the resulting cut
faces will all be rectangular but they will be different sizes.
However, looking at figure 3, we can see that cuts parallel
to the end faces will always result in cut faces which are exactly
the same size and shape. Therefore
this solid is a prism.

These
solids are not prisms



If
these solids are cut crossways several times parallel to
the face they are sitting on, the cut faces will be different.
There is no plane through which the solids can be cut to
produce uniform crosssections. 

Volume
of a prism = Area
x height
We
can find the volume of a prism by multiplying the area of the
uniform cross section by the height, where
the height is measured perpendicular to the uniform cross section.
It
is because the crosssection is uniform across the whole shape
that we can multiply the area of the crosssection by the height
of the shape.

Volume
of a rectangular prism = L
x W x H
We
could also find the volume of a rectangular prism using the
formula for prisms, A
x h.
Can you see why?

Although
by our definition a cylinder is not a prism, we can use the same formula
to calculate the volume of a cylinder as we have used for a prism
because a cylinder also has a uniform crosssection.
Example
2: Justification for the formula for the volume of a cylinder
The solids pictured below are not prisms, but we can see that they
do have a uniform crosssection. This means that if we know the area
of the base we can use the same formula to calculate the volume, V
= Area of crosssection x h
Volume
of a pyramid = 1/3 x A x h
The volume of a pyramid is 1/3 of the volume of an enclosing
prism, i.e. the volume of the red pyramid 1/3 of the volume
of the blue prism, so 1/3 of the area of the base multiplied
by its height.
This
remarkable fact also applies to cones: the volume of a cone
is one third of the volume of the enclosing cylinder.

Example
3: Find the volume of a pyramid which has a triangular base area of
5.3 cm^{2} and a height of 9 cm.
Setting
Out 
Thinking 
V
= 1/3 x A x h
V = 1/3 x 5.3 x 9
V = 15.9 cm^{3} 
I
know the area of the base of the pyramid is 5.3 cm^{2}.
The formula for the volume of a pyramid is 1/3 x area of the base
x height. Therefore the volume is 1/3 x 5.3 x 9 = 15.9 cm^{3} 
Volume
of a cylinder
We
know that the area of a circle ^{}
and therefore to find the volume of a cylinder we just need
to multiply the area of the circle by the height of the cylinder.
(Knowing
that a cylinder is a solid of uniform crosssection also means
that we know that its volume is the area of its base x height.)

Example 4: Find the capacity of a cylindrical water tank which has
a radius of 0.75 m and a height of 2.3 m.
Model 
Setting
Out 

Volume
of a cylinder
=
3.1416 x 0.75 x 0.75 x 2.3
= 5.43 m^{3}
(
I have rounded my answer to 2 decimal places)
The
units of our answer are cubic metres, but as we are interested
in the capacity of the tank it makes more sense to think of
how many litres will fit into the tank. We can convert
our answer from cubic metres to litres.
1
cubic metre = 1000 litres = 1 kilolitre
So,
5.43 cubic metres = 5.43 x 1000 litres = 5430 litres.

Volume
of a cone =
Example
5: Calculation of the volume of a cone and conversion of units.
Yuping
loves icecream! She wants to find out if she will get more icecream
in a cone compared to a small bucket. She has purchased both to perform
the test. Yuping measured the dimensions of her icecream cone and
found that it was 12 cm in height and had a radius of 4.5 cm.
If the icecream was densely packed into the cone, how much icecream
would the cone hold?
If the bucket holds 150 ml of icecream, which holds more?
Setting
Out

volume
of a cone =


=

1/3 x 3.142 x (4.5 cm)^{2} x 9 cm 
=

190.8765
cm^{3} 

Thinking

We
have found that the cone holds 190.8765 cm^{3} of icecream.
This is assuming that the icecream is very well packed into the
cone. We know that the bucket holds 150 ml of icecream. To compare
the two objects we must do so using the same units. We know that
1 cm^{3} = 1 ml, so the cone can hold approximately 190 ml of
icecream.
So if the cone can hold 190 ml of icecream and the bucket holds
150 ml of icecream which holds more? Theoretically the cone holds
more icecream but it is unlikely that the icecream would be
so tightly packed as to occupy the entire volume of the cone.
On the other hand you would expect an extra scoop of icecream
on top of the cone! 
Volume
of a sphere

A
neat relationship between the volume of a sphere and a cylinder.
Recall
that the volume of a cylinder =
If
the height of the cylinder is equal to twice the radius then
the formula for the volume can be simplified to volume =
The
volume of a sphere is
So
the volume of the sphere is 2/3 of the volume of the enclosing
cylinder.
You
can use sand and water to show this with plastic models.

Quick
quiz
1. 
What is the volume of each of these solids?


r
= 3 m


h
= 10 cm




h
= 7.5 m 

A
= 6 cm^{2} 


2. 
If the perpendicular height of the rectangular prism, pictured
below, is 8 m and the side length of its square base is 4 m, what
is its volume? 




3. 
If we know that the area of the hexagonal faces of this solid
is 9 cm^{2} and it is 5 cm thick, what is its
volume?



To
view the quiz answers, click here.
