# Pythagoras’ Theorem

Pythagoras was a Greek dude who lived about 2500
years ago. He did lots of important stuff with mathematics and philosophy, but
the thing he is probably most famous for is a simple rule he came up with about
*right-angled triangles*.

His rule simply says that if you take the length of
the *longest* side of a right-angled triangle, and square it, this should
equal the sum of the squares of the other two side lengths. Usually it’s
easier to show this with a diagram and an equation:

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_{}

The longest side in a right-angled triangle is
called the *hypotenuse*. In the diagram, the hypotenuse is side ‘a’.

### Showing Pythagoras’ Theorem working

There’s a visual way you can show how Pythagoras’
Theorem works. What you can do is take a right-angled triangle, and literally *square*
each of the side lengths – so transform the side into a square with the same
side length:

So now each side of the triangle has a square hanging
off it, with the square’s side length the same as the side length it’s
connected to. Now according to the theorem, _{}. We can see whether this is true
by dividing the squares up into little squares, and then adding them together
to work out the total area of each large square. So first, we need to divide
up the squares into grids made up of little squares, all the same size:

Now, if we add up the number of little squares in
the a^{2} area, we get 25 in total. What about the number of little
squares in the b^{2} and c^{2} areas? Well, the b^{2}
area has 16. The c^{2} area has 9. Together, there are 25 little
squares in the b^{2} and c^{2} areas. This is the same as in
the a^{2} area. If we put these ‘areas’ we’ve calculated into Pythagoras’
Theorem, we find that the theorem does in fact work.

_{}

This is a famous way of showing how Pythagoras’ Theorem works, and you should understand it and be able to talk about it in an exam. You can even draw the triangle and squares on paper, then cut them out and draw the grids, and count them yourself.

It is easiest if you draw a triangle where the
sides are 3 cm, 4 cm and 5 cm long, or any multiple of this, so 6 cm, 8 cm and
10 cm would be OK as well. If you use a grid size of 1 cm by 1 cm with these
side lengths, it means you get a nice whole number of little grid squares
inside your big squares, instead of getting little fractions of grid squares.
Counting the number of *whole* grid squares is a lot easier than trying to
add up little bits and pieces of grid squares!

Lot easier to count up complete grid squares… |
then to try and add up grid squares when some of them are not ‘whole’. |