Some equations give you a length, for example circumference of a circle. Some give you an area, for example the surface area of a sphere. And some give you a volume.
You cannot mix dimensions. For example, you cannot add a length to an area or an area to a volume.
What you need to be able to do is recognise whether an equation is giving you a length, an area, a volume or is actually total rubbish (because it is adding or subtracting different measurements).
In equations, lengths are represented by letters (variables). However, letters can also represent constants (which means that they are always the same number) and constants have no dimensions. Also, numbers have no dimensions unless you are told otherwise!
How to tell!
1. If the equation is a length then only one letter (variable) represents a length. However, lengths can be added and subtracted from each other and still give a length. For example, 10cm + 15cm is 25cm which is still a length.
2. If the equation is an area then it must be length x length. Again, it's still an area if areas are added or subtracted from each other.
3. If the equation is a volume then it must be length x length x length.
4. Be careful if they use brackets. To be safe, multiply out the brackets before making your decision.
5. Also be careful with fractions as the dimensions are allowed to cancel. So, a volume over a length will give an area, and an area over a length will give a length.
If the question on an exam paper asks you to tick boxes, only tick the ones you are sure of - Do Not Guess! This is because a wrong tick will cancel out one of the right ticks as well - sorry, that's just how they mark the papers!
If x, y, and z represent lengths and a, b and c are constants, here's some examples:
This is a volume because it's length x length x length.
This is an area because it's length x length and neither 3 nor a have any dimensions.
4ax + 3y
This is a length because it is two lengths added together.
This is a length because it is an area over a length.
Now test yourself on this idea. Below are a number of different equations.
Decide if they represent a length, area or a volume, then drag the correct anwer to the equation:
The basic thing to remember is:
Measurement is always approximate.
Whatever the measurement has been rounded to it could be half a unit either way. For example, if a sprinter is timed as doing 100 metres in 10.8 seconds his time has been rounded to the nearest tenth of a second, which means it could have been anywhere between 10.75 and 10.85 seconds.