Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want Pythagoras Theorem — finding missing sides in right-angled triangles and applying it in 3D — to become a reliable source of marks instead of a formula they only half-remember.
What query it owns: how to understand and revise Pythagoras Theorem in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the Pythagoras Theorem revision-guide angle, while Tutopiya’s Pythagoras Theorem subtopic page owns the learning resource and the free Pythagoras Theorem quiz owns the practice.

Pythagoras Theorem is one of the most frequently tested ideas in the Geometry unit of Cambridge IGCSE Mathematics (0580/0607). Whenever a question involves a right-angled triangle — on a flat diagram or inside a 3D shape — examiners expect you to recognise it quickly and apply a² + b² = c² with clear working. This guide explains exactly what the theorem covers, how to handle the question types that actually appear, and where to practise each skill.

Key takeaways

• Pythagoras Theorem applies only to right-angled triangles: the square on the hypotenuse equals the sum of the squares on the other two sides.
• The hypotenuse is always the side opposite the right angle — and it is always the longest side.
• In 3D questions, split the shape into two or more right-angled triangles and apply Pythagoras step by step.
• Always state the theorem, show substitution, and give the final length with correct units.

What is Pythagoras Theorem in Cambridge IGCSE Maths?

Pythagoras Theorem is the relationship between the three sides of a right-angled triangle. If the two shorter sides are a and b, and the hypotenuse is c, then a² + b² = c². In Cambridge IGCSE Mathematics it is used to find a missing side when two sides are known, to test whether a triangle is right-angled, and as the first step in many 3D length problems involving cuboids, prisms and pyramids.

You can read the full explanation, worked examples and notes on Tutopiya’s Pythagoras Theorem subtopic page before you attempt questions.

The core ideas you must master

These four ideas appear again and again. Learn what each one means and the exam phrasing that signals it.

Idea	What it means	How the exam uses it
Right angle	An angle of exactly 90°	“Calculate the length of AC” where angle ABC = 90°
Hypotenuse	Side opposite the right angle; longest side	”Find the hypotenuse” or the side you are not given
Missing shorter side	Rearrange to c² − a² = b²	”Work out the height of the triangle”
3D Pythagoras	Two-step use in space diagrams	”Find the space diagonal of the cuboid”

How to use Pythagoras Theorem — step by step

The safest method works for every 2D question and is the foundation for 3D problems.

1. Identify the right angle on the diagram. If it is not marked, check whether the question states “right-angled triangle”.
2. Label the hypotenuse c — the side opposite the right angle.
3. Write a² + b² = c² and substitute the two known sides.
4. Solve for the unknown. If finding a shorter side, subtract: b² = c² − a².
5. Square-root for the length. Use √ on your calculator and round only if the question asks.
6. Check: the hypotenuse must be longer than either shorter side.

Once you have worked through a few, test yourself with the free Pythagoras Theorem quiz — it tells you fast whether the method has actually stuck.

2D vs 3D: which approach does the question want?

Students lose marks by applying Pythagoras to non-right triangles or by skipping the intermediate step in 3D. Use the diagram to decide.

Situation	What to do	Typical signal words
Flat right-angled triangle	One application of a² + b² = c²	”right-angled”, right-angle symbol on diagram
Cuboid space diagonal	Pythagoras on base, then again with height	”diagonal of the cuboid”, “length AG”
Ladder against a wall	Vertical, horizontal and slant form a right triangle	”ladder”, “wall”, “ground”
Is it right-angled?	Compare c² with a² + b²	”Show that triangle ABC is right-angled”

Pythagoras Theorem in past-paper wording: command words that matter

Most lost marks in Pythagoras questions come from misreading the command word or using the wrong side as the hypotenuse. These are the command words you will see and what each one demands.

Command word / phrase	What the question wants	Typical Pythagoras stem
Calculate / Work out	Find a length, showing full method	”Work out the length of AC.”
Show that	Prove a given result — the answer is stated, so method earns marks	”Show that the length of the diagonal is 13 cm.”
Write down	State a value; minimal working (usually 1 mark)	“Write down the length of the hypotenuse.”
Give your answer correct to …	Round or give exact surd as instructed	”Give your answer correct to 3 significant figures.”
Is the triangle right-angled?	Compare a² + b² with c²	”Explain whether triangle PQR is right-angled.”

Worked exam-style stems (how to answer the wording)

Practising the wording — not just the formula — is what method marks reward. Here is how three real-style stems are answered.

1. “Triangle ABC is right-angled at B. AB = 5 cm and BC = 12 cm. Work out the length of AC.” Hypotenuse is AC. 5² + 12² = 25 + 144 = 169 → AC = √169 = 13 cm. Mark-scheme reward: correct substitution, then correct square root.
2. “A cuboid is 6 cm by 8 cm by 10 cm. Show that the length of the space diagonal is √200 cm.” Base diagonal: √(6² + 8²) = 10. Then √(10² + 10²) = √200. Reward: two clear Pythagoras steps with working — stating √200 alone scores nothing on “Show that”.
3. “A ladder of length 5 m leans against a wall. The foot of the ladder is 1.5 m from the wall. Work out how far up the wall the ladder reaches.” 1.5² + h² = 5² → h² = 25 − 2.25 = 22.75 → h ≈ 4.77 m (3 s.f.). Reward: correct rearrangement for the vertical side.

When you can recognise the wording instantly, work the full set on the Geometry topical past-paper questions and the Pythagoras Theorem quiz to lock the method in.

How Pythagoras Theorem connects to the rest of Geometry

Pythagoras feeds directly into Similarity, where scale factors link corresponding sides of similar triangles — many of which are right-angled. It also underpins trigonometry (sine, cosine and tangent ratios) and appears alongside Circle Theorems when diameters and tangents create right angles. When you are ready to mix topics, the Cambridge IGCSE Maths resource hub lets you move straight from a weak subtopic into the next.

Common mistakes students make

• Using a² + b² = c² when the triangle is not right-angled.
• Treating a shorter side as the hypotenuse because it looks longer on a tilted diagram.
• Forgetting to square-root at the end, leaving the answer as 169 instead of 13.
• In 3D, trying to jump straight to the space diagonal without finding the base diagonal first.
• Rounding too early and losing accuracy marks on multi-step questions.

When you need more support

If Pythagoras questions keep tripping you up — especially 3D diagrams — work through the Geometry topical past-paper questions and the Pythagoras Theorem quiz to pinpoint the exact gap, then get focused help from a Cambridge IGCSE Maths tutor to fix it quickly.

Frequently asked questions

Is Pythagoras Theorem hard in Cambridge IGCSE Maths? No — the formula itself is straightforward. Marks are lost when students misidentify the hypotenuse, apply the theorem to non-right triangles, or skip steps in 3D problems.

What is the quickest way to find the hypotenuse? Square both shorter sides, add the results, then square-root. The hypotenuse is always opposite the right angle.

Can Pythagoras be used in 3D? Yes. Find a right-angled triangle on the base first, then use that result as one side of a second right-angled triangle involving the height or depth.

How do I revise Pythagoras Theorem effectively? Read the subtopic notes, label diagrams carefully on every question, then take the Pythagoras Theorem quiz. Revisit any 3D problems you got wrong before moving on.

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