# Dimensions and Accuracy

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## Dimensions and Accuracy

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.**

**Tip!**

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:**

**x ^{2}y**

This is a **volume** because it's length x length x length.

**3ax ^{2}**

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.