GCSE

Pythagoras’ Theorem

The most famous equation in geometry — and the one GCSE topic almost every parent half-remembers.

What Your Child Is Learning

In any right-angled triangle, there's a relationship between the three sides:

$$a^2 + b^2 = c^2$$

where $c$ is the hypotenuse — the longest side, always opposite the right angle. The other two sides are called the "legs" or "shorter sides".

Your child uses this to find a missing side when they know the other two. It only works in right-angled triangles (for other triangles, they'll use trigonometry or the cosine rule later).

How Schools Teach It Now

The method follows three clear steps:

Step 1: Identify the hypotenuse (longest side, opposite the right angle).
Step 2: Substitute the known sides into $a^2 + b^2 = c^2$.
Step 3: Solve for the unknown side.

Finding the hypotenuse

If the two shorter sides are 6 and 8:

$$c^2 = 6^2 + 8^2 = 36 + 64 = 100$$ $$c = \sqrt{100} = 10$$

Add the squares, then square root.

Finding a shorter side

If the hypotenuse is 13 and one leg is 5:

$$a^2 = 13^2 - 5^2 = 169 - 25 = 144$$ $$a = \sqrt{144} = 12$$

Subtract the squares, then square root.

Common Mistakes to Watch For

Forgetting to square root at the end: students calculate $c^2 = 169$ and write $c = 169$ as their answer. The answer is $c = \sqrt{169} = 13$. Always check: does the answer look like a reasonable length?
Using the formula on a non-right-angled triangle: Pythagoras only works when there's a right angle. If the triangle doesn't have one, the formula gives a wrong answer — no error message, just a wrong number.
Adding when they should subtract: finding the hypotenuse = add the squares. Finding a shorter side = subtract. Mixing these up is very common. The hypotenuse is always the biggest number, so if the answer comes out bigger than the hypotenuse, something has gone wrong.

Try It Together

Worked Example

Question: A right-angled triangle has legs of 5cm and 12cm. Find the hypotenuse.

Step 1: We're finding the hypotenuse, so we add the squares.

$$c^2 = 5^2 + 12^2 = 25 + 144 = 169$$

Step 2: Square root both sides.

$$c = \sqrt{169} = 13\text{cm}$$

Check: 13 is bigger than both 5 and 12. That makes sense for a hypotenuse.

A good follow-up: "Now find the missing short side if the hypotenuse is 10 and one leg is 6." (Answer: $\sqrt{100 - 36} = \sqrt{64} = 8$)

Let Them Practise

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