Hopefully everyone reading this can state Pythagoras’ Theorem by heart – but, just in case:

“Given a right-angled triangle, the square of the length of the hypotenuse is the sum of the squares of the two shorter sides.”

Stated more simply, and with the aid of a diagram:

*c*^{2}* = **a*^{2} + *b*^{2}

But how can this result be proved? It’s no good drawing lots of right-angled triangles and measuring their sides – firstly, because it’s impossible to measure accurately; and secondly, showing that something is true in lots of cases doesn’t prove it is *always* true.

Believe it or not, there are well over 300 different proofs! Here’s one of the simplest, using the geometry of squares and triangles.

Take a look at this diagram where I have rotated the triangle (above) three times through 90°, and placed each of the four triangles so as each hypotenuse forms an inner square with side length *c*.

Let’s consider the area of the whole shape and its components.

Area of whole shape

This is

Area of the blue triangles

Each blue triangle has area given by the formula . So the total of the four triangles is .

Area of the blue triangles and yellow square combined

So we can now add together the blue and the yellow areas to get . But this is exactly the same as the area of the whole shape, so we can write:

By subtracting from both sides, we end up with q.e.d.

*(q.e.d stands for “quod erat demonstrandum”, Latin for “which was to be demonstrated”, and is the standard way to mark the end of a proof).*

Mathematicians like picking holes in proofs, and there’s one in the proof above – there’s a statement I made which needs further examination. Did you spot it? When I first drew the diagram, I said that the four hypotenuses formed an inner square. However, just because a quadrilateral has four sides the same length it doesn’t follow that it’s a square – it could be a rhombus. To satisfy ourselves that it is indeed a square, we need to show that each of its angles is 90°. Have a look at the diagram and see if you can see why.

To show each angle of the yellow quadrilateral is 90°

Look at the diagram below which shows just part of the overall shape.

Since the top triangle is right-angled, *x* + *y* = 90°. But that means *z* = 90° as well since *x*, *y *(the one in the* *bottom triangle) and *z* all lie on a straight line. This argument can be repeated for each angle of the quadrilateral, thus proving it is a square.

*In fact, you don’t need to prove that all four angles are 90° – how many would suffice?*

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