In our latest ‘Conversation Kickstarter’, Oscar investigates the famous Pythagorean Theorem.

If there’s one thing you remember from GCSE Maths, I would wager that it was Pythagoras’ Theorem. With a right-angled triangle, the squares of the opposite and adjacent sides are equal to the square of the hypotenuse.

But where did it come from? Why does it work? Where is it used? In this short lesson, we’re going to walk through the origins of this famous theorem, its proof and why we couldn’t live without it.

## Pythagoras: The Man, The Myth, The Legend?

While the theorem might be named after the Greek philosopher Pythagoras, it’s entirely plausible that it was actually discovered thousands of years prior to his birth by Babylonian mathematicians. In fact, there is even evidence that it appeared in China and India before Pythagoras was able to prove it.

Pythagoras’ life is full of mystery. Around 530 BC, he travelled over to Croton in Italy where he founded a small community consisting of his followers who would adhere to his philosophical views. Dubbed, unsurprisingly, as Pythagoreanism, this way of life involved strict rules – some of which were truly bizarre. For instance, new members of the community were not allowed to meet Pythagoras until they had undergone a five year initiation, during which they were forced to remain silent!

Even weirder, Pythagoras declared that the consumption of beans, yes beans, was prohibited and those who failed to follow the ‘laws’ of the commune would be expelled. Once banished, the remaining members would create a tombstone for the departed member. When Pythagoras ‘discovered’ his famous theorem, it is said that he sacrificed an ox in sheer joy. Creepy stuff. Anyway, that’s quite enough on that front – how does the theorem work?

## Counting Squares

There are several ways of ‘proving’ Pythagoras’ Theorem, we’ll take a look at a really simple, but rather clever proof. First, let’s just draw a right-angled triangle:

Now, when we ‘square’ a side, we can imagine a literal square being formed using that edge of the triangle. We can do this for all three sides of the triangle:

Intuitively, we can see that the sum of the area of A and B will be equal to the area of C. For example, if side a = 3cm , b = 4cm and c = 5cm, then the square A = 3 x 3 = 9cm sq, B = 16cm sq and C = 25cm sq. Sure enough, the Pythagorean Theorem holds as 9 + 16 = 25 (or, a^2 + b^2 = c^2 - yes, Wix doesn't have equation editing functionality, I'm sure we'll live). However, this isn’t enough for a general proof, we need to show that it is always true. Let’s construct a few more triangles around the large square C:

Now we have a large square (let’s call it alpha, α), with sides (a + b) and an area of (a + b)^2. The area of α can also be thought of as the sum of the areas of those four triangles and the tilted square C. Remembering that the area of a right-angled triangle is ½ (a x b), α = 4[ ½(a x b)] + c^2.

Alright, let’s tidy all of this up. With basic algebra, we can simplify (a x b) to just (ab) so this ugly thing 4[ ½(a x b)] can be expressed as 2(ab), so now we have α = 2ab + c^2. We said earlier that the area of the big square α can also be shown to be (a + b)^2, this means we substitute that in for α, giving us: (a + b)^2 = 2ab + c^2.

We’re almost there! We can expand the brackets (a + b)^2, doing so leaves us with a2 + b2 + 2ab. Putting that back into the above equation gives us a^2 + b^2 + 2ab = 2ab + c^2. Both sides have the term 2ab, so we can subtract 2ab from the equation and that leaves us with the famous Pythagorean Theorem, a^2 + b^2 = c^2. Do take some time to go over the proof again, once you understand it, it’s really quite elegant! Well, that or I’m a little weird…

## Putting Pythagoras to Work:

So we’ve spoken about who Pythagoras was and how the theorem works, but where can we apply it? Students often mutter the immortal words “but how can I use this in the real world?”, so, here’s a few places where the Pythagorean Theorem comes in handy!

An obvious application is construction and engineering. If you are building a sloped roof and want to find out (given the width and height of the roof) the diagonal length of the roof, you are going to need Pythagoras’ Theorem. Given any two lengths and a right-angled triangle, you’ll always be able to find the missing side with the theorem.

Better yet, you can also use it to make sure a square or rectangle has 90 degrees angles at all sides. First, diagonally cut the square in two halves to make two right-angled triangles, then you can measure all three sides of one of these triangles. If a^2 + b^2 = c^2 holds, you can be sure that you have a perfect square! This is especially useful if you’re building the foundations of a new house and want to ensure the sides are all at right-angles.

The other common application of Pythagorean Theorem is in navigation. Let’s say I’m out at sea and the port I want to head to is 100 miles north and 200 miles east. Working through the equation, 100^2 + 200^2 = 50,000. Then I can find ‘c’ (the distance) through the square root of 50,000 miles, which is roughly 224 miles.

If we were feeling even more adventurous, we could find the angle from our position using tan(θ) = 200/100, so our angle would be tan^-1(200/100) ≈ 63 degrees. Not bad! This is used frequently by captains to find their bearings and locate the distance to their destination. If that last part came out of the blue for you, now’s the perfect time to practise some basic trigonometry to help you find angles in a right-angled triangle.

Well, I think that brings things to a natural conclusion. It’s easy to lose the forest for the trees when you’re cramming in information during class, we often miss out the wider context of what we’re actually learning! I hope this blog has shed some light on the Pythagorean Theorem and refreshed some memories!

You know the drill, pay it forward and see who else might be interested in the origins and use of this famous old theorem. Bring it up over dinner or give your friend a ring; let me know on our Facebook or Twitter channels if you were able to 'kickstart' a conversation!

Blog Post Crafted by Oscar

Oscar studies Philosophy, Politics and Economics (PPE) at the University of Warwick.

When he's not studying PPE or tutoring GCSE Maths and Science, Oscar plays the saxophone and is the Musical Director for the University of Warwick Big Band.

In 2017 he set up his own jazz function band, Mirage Quartet, and has been a keen collaborator and ambassador for Bromley Youth Music Trust.