Volumes
Dimensions

Most quantities are based on three fundamental quantities, mass, length and time. The letters M, L and T are used to describe the dimensions of these quantities.

If m is the mass of an object, then we write

[m] = M

to show that M is the dimension of m.

If x is a length, then, similarly,

[x] = L

[x2] = L2

[x3] = L3

If t is a time, then

[t] = T

Here the emphasis will be on quantities involving lengths.

As areas are calculated by multiplying two lengths,

[Area] = L2   or   Dimensions of Area = L2

Similarly,

[Volume] = L3   or   Dimensions of Volume = L3

The ideas of dimensions can be used to check formulae.

Worked Examples

1

The area of a circle is given by

A = πr2

(a)

What are the dimensions of A and r?

Show me

A is an area, so [A] = L2

r is a length, so [r] = L and [r2] = L2
(b)

What are the dimensions of π?

Show me

The dimension of the formula A = πr2 must be consistent, so

[A] = [π] × [r]2

But using the results from (a),

L2 = [π] × L2
[π] = = L0
= 1

The number π has no dimension.

Note

Numbers that are not lengths or other quantities have no dimension.

2

If x, y and z are all lengths, state whether each expression below could be for a length, an area or a volume.

(a)

x + y + z

Show me
[x + y + z] = [x] + [y] + [z]
= L + L + L

As the formula contains 3 terms, all of dimension L, this expression gives a length.
(b)

xyz

Show me
[xyz] = [x] × [y] × [z]
= L × L × L
= L3

As the dimension of xyz is L3 this expression gives a volume.
(c)

+ y

Show me
+ [y] = + L
= 1 + L

The first term has no dimension and the second has a dimension, L, so the expression is not dimensionally consistent.
(d)

Show me
=
=
=
= L

As the dimension of this quantity is L, this expression would give a length.
(e)

xy + xz + yz

Show me
[xy + xz + yz] = [xy] + [xz] + [yz]
= L2 + L2 + L2

So the expression contains three terms, all of which have dimension L2, and so could represent an area.

Challenge

A man wishes to take 4 litres of water out of a large tank of water, but he has only one 5-litre and one 3-litre jar. How can he do it?

Exercises

If x, y and z all represent lengths, consider each expression and decide if it could be for a length, an area, a volume or none of these.

(a)
xy
(b)
(c)
(d)
(e)
(f)
+
(g)
+ y3
(h)
(i)
2xyz

Which of the formulae listed below are not dimensionally consistent, if p, q and r are lengths and A is an area?

(a)
p = q + r
(b)
p = + r
(c)
A = pqr
(d)
A =
(e)
p =
(f)
q2 = p2 + r2

The diagram shows a triangular wedge. Use dimensions to select the formula from the list below that could give the surface area of the wedge.

A = abc + bd + ac
A = ab + ac + bc + cd
A = a + b + c + d
A = abc + bcd + abd

Consider sinθ =

What are the dimensions of sinθ?

sinθ has dimension L

The diagram shows a hole, which has a radius of r at the surface and a depth d.

Which of the following could give the volume of the hole?

V = πrd
V = πr2d
V = π2rd
V = πr3d

By considering dimensions, decide whether the following expressions could be a formula for

perimeter, area, or volume.

In the expressions below, a, b and c are all lengths.

(a)
a + b + c
(b)
πa3 + πa2b

The diagram shows a child's play brick in the shape of a prism.

The following formulae represent certain quantities connected with this prism.

Which of them represent areas?

πab π(a + b) πabl π(a + b)l

One of the formulae in the list below can be used to calculate the area of material needed to make the curved surface of the lampshade in the diagram.

(i)πh(a + b)2

(ii)πh2(a + b)

(iii)πh(a + b)

(iv)πh2(a + b)2

State which formula is correct.

Information

Archimedes (287BC-212BC), a Greek Mathematician, was once entrusted with the task of finding out whether the King's crown was made of pure gold. While taking his bath, he came up with a solution and was so excited that he dashed out into the street naked shouting "Eureka" (I have found it). The container that you use in the Science laboratory to measure the volume of an irregular object is known as an Eureka can (named after this incident). Archimedes was so engrossed in his work that when his country was conquered by the Romans, he was still working hard at his mathematics. When a Roman soldier ordered him to leave his desk, Archimedes replied, "Don't disturb my circles." He was killed by that soldier for disobeying orders.

Archimedes' greatest contribution was the discovery that the volume of a sphere is that of a cylinder whose diameter is the same as the diameter of the sphere. At his request, the sphere in the cylinder diagram was engraved on his tombstone.