Length, area and volume: adding

A 1 dimensional line, a 2 dimensional square, and a 3 dimensional cubeLengths, areas and volumes cannot be added together

Think about this question – can you add a length to an area?

Hopefully you have come to the conclusion that this is in fact impossible. The reason for this is that length and area have different units, and quantities with different units cannot be added together.

This also explains why we cannot add a time to a weight or why we cannot simplify 3a + 2b, as the answer would be nonsense!

One final consideration is that we cannot add or subtract dimensionless quantities (normal numbers such as 5, 12.8 and \frac{3}{4}) to a quantity with dimensions.

While quantities with different units cannot be added – they can often be multiplied.

You will need to know the following for your exam:

  • Number + Length (not allowed)
  • Number + Area (not allowed)
  • Number + Volume (not allowed)

  • Length + Area (not allowed)
  • Length + Volume (not allowed)
  • Area + Volume (not allowed)

  • Length + Length = Length
  • Area + Area = Area
  • Volume + Volume = Volume

Question

{a} is a length. Does the following calculation make sense?

{a}~+~{12}

No, the number 12 has no units while {a} would be measured in metres. A quantity with units cannot be added to a unit-less quantity.

Question

{a} and {b} are both lengths.

Does the following calculation make sense?

3 {a}~+~{b}

Yes, as {a} and {b} are both lengths they can be added as normal.

Question

Assad looks on the internet to try to find an equation for the area of a kite.

He finds a post on a web forum that says:

5 {a}~–~\frac{1}{2}~{b^2} is the equation for the area of a kite, where {a} is the length of the base and {b} is the height.

Explain why Assad knows this is not the correct equation.

It cannot be correct as {a} is a length and {b^2} is an area. A length and an area cannot be added together.

This is because lengths would be measured in metres ( {m}), and areas are measured in metres squared ( {m^2}). They are two different units.