Pythagoras' Theorem Explained with Examples

Pythagoras' Theorem is one of the most famous results in all of mathematics, and for good reason. It lets you find the missing side of a right-angled triangle when you know the other two sides. You will use it constantly from Year 8 onward, in trigonometry, coordinate geometry, measurement and even physics.

This guide explains what the theorem says, why it works, and how to apply it step by step.

What the theorem says

In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Written as a formula:

c² = a² + b²

Here, c is the hypotenuse (the longest side, always opposite the right angle) and a and b are the other two sides.

This only works for right-angled triangles. If there is no right angle, you need different tools (like the cosine rule, which you will meet later).

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Finding the hypotenuse

When you know the two shorter sides and need to find the longest side, use the formula directly.

Example 1

A right-angled triangle has shorter sides of 3 cm and 4 cm. Find the hypotenuse.

c² = 3² + 4²

c² = 9 + 16

c² = 25

c = 5 cm

This is the classic 3-4-5 triangle. You will see it everywhere in maths problems.

Example 2

A right-angled triangle has shorter sides of 5 m and 12 m. Find the hypotenuse.

c² = 5² + 12²

c² = 25 + 144

c² = 169

c = 13 m

Another well-known set: 5-12-13.

Example 3

A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the hypotenuse.

c² = 6² + 8²

c² = 36 + 64

c² = 100

c = 10 cm

Notice this is just the 3-4-5 triangle doubled. Pythagorean triples can be scaled up.

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Finding a shorter side

Sometimes you know the hypotenuse and one shorter side, and need to find the other. Rearrange the formula:

a² = c² - b²

Example 4

A right-angled triangle has a hypotenuse of 10 cm and one shorter side of 6 cm. Find the other side.

a² = 10² - 6²

a² = 100 - 36

a² = 64

a = 8 cm

Example 5 (with a surd answer)

A right-angled triangle has a hypotenuse of 7 cm and one shorter side of 4 cm. Find the other side.

a² = 7² - 4²

a² = 49 - 16

a² = 33

a = √33 which is approximately 5.74 cm

Not every answer will be a whole number. When it is not a perfect square, leave it as a surd (square root) for exact answers, or round to a sensible number of decimal places if the question asks for an approximation.

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Common Pythagorean triples

These sets of whole numbers satisfy Pythagoras' Theorem. Knowing them saves time in tests.

a	b	c
3	4	5
5	12	13
8	15	17
7	24	25

Any multiple of these also works. For example, 6-8-10 and 9-12-15 are both multiples of 3-4-5.

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Real-world applications

Pythagoras' Theorem is not just a textbook exercise. Here are situations where it comes up in real life:

• Finding the distance between two points on a map or grid (this leads directly into coordinate geometry)
• Checking if a corner is square: Builders use the 3-4-5 rule to check right angles on construction sites
• Calculating the length of a ladder needed to reach a certain height on a wall
• Working out the diagonal of a rectangle (such as the screen size of a TV or monitor)

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Common mistakes to avoid

• Using the formula on a triangle that is not right-angled: Always check for that right angle first.
• Forgetting to square root at the end: c² = 25 does not mean c = 25. You need to take the square root, so c = 5.
• Subtracting the wrong way when finding a shorter side: It must be the hypotenuse squared minus the known side squared, not the other way around. If you get a negative number under the square root, you have subtracted in the wrong order.
• Mixing up which side is the hypotenuse: It is always the longest side, directly opposite the right angle.

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Where to from here

Once you are comfortable with Pythagoras' Theorem, you are well placed to tackle trigonometry (sin, cos, tan), which builds on the same right-angled triangle setup. You will also use it in coordinate geometry to find distances between points on the Cartesian plane.

If you want to practise applying Pythagoras' Theorem with guided feedback, imSteyn covers it as part of the Year 8 and Year 9 curriculum. Rather than just telling you whether an answer is right or wrong, it walks you through the reasoning so you understand each step.

The beauty of Pythagoras' Theorem is its simplicity. Three sides, one equation, and an enormous number of practical uses. Master it now and you will keep coming back to it for years.