Pythagoras' theorem is one of the most reliably tested topics in GCSE Maths — it appears on nearly every paper across AQA, Edexcel and OCR. Once you understand it properly, it is also one of the most consistent mark-earners you can have. This guide covers everything from the basics through to 3D Pythagoras on Higher tier.
What Is Pythagoras' Theorem?
Pythagoras' theorem states that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
$$a^2 + b^2 = c^2$$
Where c is the hypotenuse — the longest side, always opposite the right angle.
This formula is not given on the AQA, Edexcel or OCR formula sheet. You must know it from memory.
Identifying the Hypotenuse
Before you use the theorem, you must correctly identify the hypotenuse:
- It is always opposite the right angle (the 90° corner, shown by a small square)
- It is always the longest side
- It is always the side you label c
If you label the wrong side as c, the entire calculation is wrong. Take 5 seconds to identify it before writing anything.
Method 1: Finding the Hypotenuse
Use this when the two shorter sides are given and you need the longest side.
Formula: c² = a² + b², then take the square root.
Worked Example 1
Find the hypotenuse of a right-angled triangle with legs 6 cm and 8 cm.
Step 1: Write the formula: c² = a² + b²
Step 2: Substitute: c² = 6² + 8² = 36 + 64 = 100
Step 3: Square root: c = √100 = 10 cm
This is a 3-4-5 Pythagorean triple scaled by 2. Memorising common triples (3-4-5, 5-12-13, 8-15-17) can save time on non-calculator papers.
Worked Example 2
A ladder is leaning against a wall. The foot of the ladder is 1.8 m from the base of the wall and the ladder reaches 4.2 m up the wall. How long is the ladder?
Step 1: Identify the triangle. The ladder is the hypotenuse, the wall and ground are the two legs.
Step 2: c² = 1.8² + 4.2² = 3.24 + 17.64 = 20.88
Step 3: c = √20.88 = 4.57 m (to 3 s.f.)
Method 2: Finding a Shorter Side
Use this when the hypotenuse and one shorter side are given and you need the other shorter side.
Formula: a² = c² − b², then take the square root.
Worked Example 3
A right-angled triangle has a hypotenuse of 13 cm and one leg of 5 cm. Find the other leg.
Step 1: a² = c² − b² = 13² − 5² = 169 − 25 = 144
Step 2: a = √144 = 12 cm
(This is the 5-12-13 triple — worth recognising.)
Worked Example 4
Find the perpendicular height of an isosceles triangle with equal sides of 10 cm and a base of 12 cm.
Step 1: The perpendicular height bisects the base. This creates a right-angled triangle with hypotenuse 10 cm and base leg 6 cm (half of 12).
Step 2: height² = 10² − 6² = 100 − 36 = 64
Step 3: height = √64 = 8 cm
Checking If a Triangle Is Right-Angled
Sometimes a GCSE question asks you to determine whether a triangle is right-angled.
Method: If a² + b² = c² (where c is the longest side), the triangle is right-angled. If they are not equal, it is not.
Worked Example 5
Is a triangle with sides 7 cm, 24 cm and 25 cm right-angled?
Check: 7² + 24² = 49 + 576 = 625. And 25² = 625. ✓
Yes, it is right-angled (this is the 7-24-25 Pythagorean triple).
Common Mistakes to Avoid
Mistake 1: Squaring instead of square rooting at the end. Students write c² = 100 and then give the answer as 100. The answer is c = √100 = 10. Always take the square root.
Mistake 2: Adding when finding a shorter side. When finding a leg (not the hypotenuse), you subtract: a² = c² − b². Using + instead gives a completely wrong answer.
Mistake 3: Rounding too early. If a question has multiple steps, keep the exact value (e.g. √50) through intermediate calculations and round only at the final step.
Mistake 4: Not identifying the hypotenuse first. Rushing in without identifying which side is c causes the wrong structure. Always mark the right angle and identify the hypotenuse before writing the formula.
Pythagorean Triples Worth Knowing
A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c². These often appear on non-calculator papers because they give exact answers.
| Triple | Scaled versions you might see |
|---|---|
| 3, 4, 5 | 6, 8, 10 / 9, 12, 15 / 15, 20, 25 |
| 5, 12, 13 | 10, 24, 26 |
| 8, 15, 17 | — |
| 7, 24, 25 | — |
If you spot numbers from a triple in a question, the answer will be exact — no rounding needed.
Higher Tier: 3D Pythagoras
On Higher tier, Pythagoras is extended into three dimensions. The method is the same — you apply the theorem twice, using a diagonal across a 3D shape.
Worked Example 6 — Space Diagonal of a Cuboid
A box (cuboid) measures 5 cm × 4 cm × 3 cm. Find the length of the longest diagonal.
Step 1: Find the diagonal across the base first. d₁² = 5² + 4² = 25 + 16 = 41 → d₁ = √41 cm
Step 2: Now use d₁ and the height to find the space diagonal. d² = (√41)² + 3² = 41 + 9 = 50
Step 3: d = √50 = 5√2 ≈ 7.07 cm (to 3 s.f.)
Key insight: You can also do this in one step: $$d = \sqrt{l^2 + w^2 + h^2} = \sqrt{25 + 16 + 9} = \sqrt{50}$$
Worked Example 7 — Height of a Pyramid
A square-based pyramid has a base of 10 cm × 10 cm and slant edges of 13 cm. Find the perpendicular height.
Step 1: Find the distance from the centre of the base to a corner. Half-diagonal = √(5² + 5²) = √50 = 5√2
Step 2: Apply Pythagoras with the slant edge as hypotenuse. height² = 13² − (5√2)² = 169 − 50 = 119
Step 3: height = √119 = 10.9 cm (to 3 s.f.)
Pythagoras in Coordinates (Higher)
The distance between two points (x₁, y₁) and (x₂, y₂) is found using Pythagoras:
$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$
Example: Find the distance between points A(1, 3) and B(7, 11).
$$d = \sqrt{(7-1)^2 + (11-3)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$
How Pythagoras Links to Other Topics
Pythagoras rarely appears in isolation on exam papers. It is commonly combined with:
- Trigonometry: Finding a side using Pythagoras, then using it in a trig ratio. See our trigonometry guide.
- Area: Using Pythagoras to find the height of a triangle, then calculating area = ½bh.
- Vectors (Higher): Finding the magnitude of a vector using Pythagoras.
- Circle problems: Finding the chord length or distance from centre to chord using Pythagoras.
When you see a right angle in any question — even if Pythagoras is not explicitly mentioned — it is a signal to consider whether the theorem applies.
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