Way way back many centuries ago, Pythagoras and his students stated the following theorem, which is still used today.

**In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.**

What does this mean mathematically? In the right-angled triangle below a^{2} + b^{2} = c^{2}

Sounds really complicated? Do not fear! We'll break it down and you'll see it's not so bad.

1) So, step 1 is to remember that this **only **works with **right-angled** triangles! If you can't see that one angle = 90° then this theorem will not work.

2) What is the **hypotenuse**? The hypotenuse is always the **longest side** and the one **opposite** the right angle. In the triangle above the hypotenuse is length C.

3) What does a ² + b² = c² ? It means that if we square the length of **side a** (multiply it by itself) and **add** it to the square of **side b **(multiply it by itself) then the answer will be the same as the square of **side c**.

Have a look at the image below, and this may make a bit more sense...

If we count the number of squares in **square** **a (3 × 3)** and then add that to the number of squares in **square** **b (4 ×: 4) **we have **9 + 16 = 25**

Now, if this theorem is correct this total should be the same as the number of squares in **square** **c**... and it is! There are 25 squares in **square c**.

4) So, now we know the square of side C, how do we find the actual length? From here, we need to find the square root of 25, which we know is 5

(The symbol above is the square root symbol on a calculator. We could work this one out, but if the numbers get more complicated you can use your calculator.)

**Example**

Here's an example where you will need to use your calculator to work out the length of side c.

Use Pythagoras' Theorem to calculate the length of the hypotenuse, c, in the following right-angled triangle. Give your answer to 3 significant figures.

**Answer**

By Pythagoras' Theorem,

c^{2 }= 3^{2} + 8^{2}

c^{2} = 9 + 64

c^{2} = 73

c = √73

c =** 8.54 **(to 3 significant figures)

**Reminder** - 3 significant figures mean that we leave our answers as the first 3 numbers that we see. We always need to look to the 4th number when we do this. If the 4th number is a 5 or higher then we round up the third number. For example, if the answer was 8.546, we would write this as 8.55 (3.s.f)