Pythagoras’ Theorem

Pythagoras was a Greek mathematician who lived around 2500 years ago. He’s best remembered for the theorem that was named after him – Pythagoras’ Theorem! We’ll look at why it’s named after him later, but first, how do we use it?

Portrait of Pythagoras

Pythagoras’ theorem is used to find missing sides in right-angled triangles. We assign the value c to the longest side, or hypotenuse, of a right-angled triangle. We assign the values a and b to the remaining sides.

Right-angled triangle

For any right-angled triangle labelled this way, it is true that a2 + b2 = c2. Jump to here if you want to see how the theorem can be proved.

This allows us to find the length of the third side when we know the other two.

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Right-angled triangle with sides of 5cm, 8cm and x

In this triangle, a = 8cm, b = 5cm and c is represented by the letter x on the diagram. To find the length of the side marked x, we substitute into the formula, to give:

  • 82 + 52 = x2

Evaluating the left-hand side gives:

  • 89 = x2

Square rooting both sides gives:

  • x = 9.43cm (2d.p.)

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Right-angled triangle with sides of 62mm, 8.3cm and a

In this triangle, b = 62mm, c = 8.3cm and a is the side we are trying to find.

The first step here is to convert the given sides to the same units. We will change c = 8.3cm = 83mm.

We can now substitute into the formula:

  • a2 + 622 = 832

Evaluating the squares gives:

  • a2 + 3844= 6889

Subtracting 3844 from both sides gives:

  • a2 = 3045

Finally, we square root:

  • a = 55.1mm (1d.p.)

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Using Pythagoras’ Theorem, we can also find the length of a diagonal of a square. Let’s use the unit square – that is a square with sides of 1 unit (1cm, 1inch, 1 mile – it doesn’t really matter).

Split square with right-angled triangles

If we split the square in half, we have a right-angled triangle with sides a = b = 1

We can find the length of the hypotenuse (the diagonal) by using Pythagoras’ theorem.

This means:

  • 12 + 12 = c2
  • 12 = 1 so we have:
  • c2 = 2

Square rooting both sides gives c = √2. This number is an irrational number, which means it goes on forever and can’t be written as a fraction. It is roughly equal to 1.414.

Isosceles Triangles

The height of an isosceles triangle can also be found using Pythagoras’ theorem. Consider a triangle with base x and two equal sides measuring y:

Isosceles triangle

We can bisect the triangle, or split it into two equal parts. This is shown by the dotted line which will meet the base at a right angle.

By doing this, we create two right-angled triangles! The base of one will be (x ÷ 2) and the hypotenuse will be y. We can use these to find the height of the isosceles triangle.

Let’s try it with some numbers rather than letters.

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Isosceles triangle with sides of 5cm, 6cm and 5cm

Here, our base is 6cm and our two sides are 5cm. Bisecting the triangle will give a base of 3cm and a hypotenuse of 5cm. This gives us the following right-angled triangle.

Right-angle triangle with sides of 5cm and 3cm

Using Pythagoras’ theorem with a = 3cm and c = 5cm we have:

32 + b2 = 52

Evaluating the squares gives:

9 + b2 = 25

Subtracting 9 from both sides gives:

b2 = 16

Finally, we square root to get:

b = 4cm

This answer is different to the ones we had seen before. This isn’t because it’s an isosceles triangle; it’s because 3, 4 and 5 belong to a special group of numbers. Which brings us to…

Pythagorean Triples

What’s special about 3rd April 2005?

It was a Pythagorean day of course! The date 03/04/05 has three numbers that make up a Pythagorean triple.

That means that the numbers satisfy Pythagoras’ theorem:

  • 32 + 42 = 52

There are an infinite number of Pythagorean triple. If 3, 4 and 5 are a Pythagorean triple then so are 6, 8 and 10 or 9, 12 and 15. If you multiply them all by the same number you will get a new set of triples.

Let’s talk about primitive Pythagorean triples which are Pythagorean triples that aren’t a multiple of another triple. There are only 16 of these below 100. We’ve looked at one (3, 4 and 5) but see if you can find another! The full list is at the bottom – no cheating!

When will be the next Pythagorean day?

Well, we have to wait till 2025 to get the next primitive Pythagorean day – it will fall on 24th July 2025 (24, 7, 25) and that will be the last one in this century. Don’t worry though – there are a few non-primitive Pythagorean day’s coming up; the next one is on 16th December 2020 so mark your calendars and don’t forget to say HAPPY PYTHAGORAS!

Where Did This Theorem Come From?

Pythagorean triples have been known about since ancient times. Plimpton 322 is a boring name for a very exciting object, and a relatively famous mathematical artefact. It dates from around 1800BC and has a table of Pythagorean triples on it.

Plimpton 322

Don’t worry if you don’t recognise the numbers; they’re written in cuneiform. Plimpton 322 was written by the Babylonians – a group of people who lived close to what is now Iraq.

How can that be? Pythagorean triples existed over a 1000 years before Pythagoras was even born!?

Well, to put it simply, Pythagoras didn’t discover the relationship between the sides of a right-angled triangle. He just wrote it down, proved it and named it after himself.

In fact, about 200 years before Pythagoras proved the theorem, a Hindu mathematician called Baudhayana developed it and wrote it in his book Baudhayana Sulba Sutra.

How Do You Prove It?

It’s important to understand there’s a difference between showing something and proving it.

If you show something, it means you’ve said it works for a particular set of values or you’ve used a diagram or demonstration to make it clearer.

Proving it is more complicated; it means checking that it will work for every possible scenario so, in this case, every right-angled triangle that can be drawn or imagined.

There are a few different ways to show that Pythagoras’ theorem works. The most commonly used one is by drawing squares on each side of a triangle, as shown below:

Proving Pythagoras' Theorem using squares

If this is drawn and measured accurately, you can find the areas and check that the purple square has the same area as the red and blue squares combined.

Why not try it on some squared paper? Make sure the angle is a right angle and that you measure each side carefully.

Very similar to this is the Pythagoras water video:

The easiest way to prove Pythagoras’ theorem is proof by rearrangement – a proof often attributed to Pythagoras.

Consider four identical right-angled triangles. We label the hypotenuse c and the remaining two sides a and b:

Four identical right-angled triangles

We can arrange these four triangles in a square, as shown. The side length of this square is a + b . The red area in the centre is also a square and has area c2

Four triangles in a square

We can also arrange the triangles into two rectangles. The side length of this square is also a + b. The area of the smaller white square is a2 and the area of the larger white square is b2.

Triangles in rectangles

Since the area of the triangles hasn’t changed and the area of the large square hasn’t changed, the area of the two white squares must equal the area of the red square. In other words:

  • a2 + b2 = c2

So, there you have it! Bundles of information on Pythagoras and his theorem… that wasn’t really his to start with.

Pythagorean Triples Answers

(3, 4, 5)(5, 12, 13)(8, 15, 17)(7, 24, 25)
(20, 21, 29)(12, 35, 37)(9, 40, 41)(28, 45, 53)
(11, 60, 61)(16, 63, 65)(33, 56, 65)(48, 55, 73)
(13, 84, 85)(36, 77, 85)(39, 80, 89)(65, 72, 97)

Resource Links on Pythagoras’ Theorem

Pythagoras’ Theorem in 2D Shapes Lesson 1: An Introduction

Lesson pack on Pythagoras' Theorem

Pythagoras’ Theorem: Choose Your Own Adventure

Choose your own adventure lesson resource on Pythagoras' Theorem

Pythagoras’ Theorem: KS3 Walkthrough Worksheet

Pythagoras' Theorem worksheet

Pythagoras’ Theorem in 3D Shapes

Pythagoras' Theorem in 3D Shapes

Maths Mastery: Pythagoras’ Theorem – KS3 Maths

Maths Mastery on Pythagoras' Theorem

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