Pythagoras Theorem: A Key KS3 Maths Topic for GCSE
Pythagoras’ Theorem is one of the most fundamental and essential topics in KS3 Mathematics. It lays the foundation for understanding geometry, trigonometry, and even GCSE problem-solving questions involving right-angled triangles.
You’ll encounter it in topics such as:
- Finding missing sides of triangles
- Applying to coordinate geometry
- Real-life and exam-style word problems
So, mastering Pythagoras’ Theorem early in KS3 helps students build a strong base for GCSE success!
Page Contents
What Is Pythagoras Theorem?
Pythagoras’ Theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the other two sides.
$$a^2 + b^2 = c^2$$
where:
- a and b are the shorter sides
- c is the hypotenuse
In words: The square on the hypotenuse is equal to the sum of the squares on the other two sides.
How to Use Pythagoras Theorem
You can use this formula in two main ways:
(a) To find the hypotenuse
If you know both shorter sides a and b:
$$c = \sqrt{a^2 + b^2}$$
(b) To find a missing shorter side
If you know the hypotenuse c and one shorter side a:
$$b = \sqrt{c^2 – a^2}$$
Tip: Pythagoras only works for right-angled triangles! Always check the question carefully before applying it.
Examples
Example 1: Finding the Hypotenuse
A right-angled triangle has shorter sides \(a = 3\) cm and \(b = 4\) cm. Find the hypotenuse \(c\).
Write down the formula and apply the numbers:
$$c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$
Answer: The hypotenuse \(c = 5\) cm.
Example 2: Finding a Missing Side
In a right-angled triangle, the hypotenuse is \(c = 13\) cm and one shorter side is \(a = 5\) cm. Find the other side \(b\).
Write down the formula and apply the numbers:
$$b = \sqrt{13^2 – 5^2} = \sqrt{169 – 25} = \sqrt{144} = 12$$
Answer: The missing side \(b = 12\) cm.
Example 3: Real-Life Word Problem
A ladder leans against a wall. The ladder is \(10\,\text{m}\) long, and the foot of the ladder is \(6\,\text{m}\) away from the wall. How high does the ladder reach up the wall?
Let \(c = 10\) (ladder), \(a = 6\) (distance away from the wall), and \(b\) be the height.
$$b = \sqrt{10^2 – 6^2} = \sqrt{100 – 36} = \sqrt{64} = 8$$
Answer: The ladder reaches \(8\,\text{m}\) high on the wall.
Common Mistakes to Avoid
- Forgetting that Pythagoras’ Theorem only applies to right-angled triangles.
- Mixing up which side is the hypotenuse (it’s always the side opposite the right angle).
- Forgetting to square and then square root (e.g. trying to just add sides directly).
- Rounding too early — keep answers in exact form (√) unless the question asks for decimals.
Examiner Tips & Tricks
- Always label sides clearly before substituting into the formula.
- Check that your final answer makes sense — the hypotenuse should always be the longest side!
- When the question involves a 3D shape, apply Pythagoras step-by-step (e.g. base diagonal first, then the height).
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