Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Pythagoras' Theorem

In one sentence. a² + b² = c² — finding missing sides in right-angled triangles, and spotting the hidden right triangles inside other shapes.

Trigonometry — one of 4 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Euclid. Learn Pythagoras' Theorem, teach it, then defend it against Euclid.

Where this sits in the specification

Specification 4MA1, 4.8 Pythagoras' theorem. Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 4.8 — Pythagoras' theorem

1 · Understand it

What it is

In any RIGHT-ANGLED triangle, the square on the hypotenuse (the side opposite the right angle — always the longest) equals the sum of the squares on the other two sides: a² + b² = c². Finding the hypotenuse: add the squares, root. Finding a leg: subtract the squares, root.

Why it works

Tilt a square of side c inside a square of side (a + b): the leftover space is four copies of the triangle, and the algebra of the areas forces c² = a² + b². It's a fact about AREAS — the squares are real squares, drawn on the sides — which is why the theorem talks in squares at all.

When a question is asking for it

Two sides of a right triangle known, third wanted. But mostly: hidden right triangles — diagonals of rectangles, chords with perpendiculars, isosceles triangles cut down the middle, distances between coordinates, ladders on walls. If you can DRAW a right angle into the figure, Pythagoras is available.

How to hold it in your head

Ask one question first, every time: is the missing side the HYPOTENUSE (longest, opposite the right angle) or a LEG? Hypotenuse → squares add. Leg → squares subtract. The answer must be longer than either leg if it's the hypotenuse, shorter than the hypotenuse if it's a leg — a built-in sanity check.

2 · The mistakes that cost marks

  • Adding the squares regardless. Finding a LEG subtracts: a² = c² − b².
  • Forgetting the final square root and reporting c².
  • Applying it to a triangle with no right angle. The theorem is exclusive to right triangles (the cosine rule generalises it).
  • Calling the longest-LOOKING side the hypotenuse. It's the side opposite the right angle — check the diagram, not the drawing scale.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

A right-angled triangle has legs 7 cm and 24 cm. Find the hypotenuse.

  1. Hypotenuse wanted → add squares: 7² + 24² = 49 + 576 = 625.
  2. c = √625 = 25 cm.

Answer: 25 cm

An isosceles triangle has base 16 cm and equal sides 17 cm. Find its height.

  1. The height splits it into two right triangles: half-base 8, hypotenuse 17.
  2. h² = 17² − 8² = 289 − 64 = 225.
  3. h = 15 cm.

Answer: 15 cm

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — A right-angled triangle has legs of 7 cm and 24 cm. Work out the length of the hypotenuse. (answer in cm)
  2. Grade 7 — A right-angled triangle has hypotenuse 13 cm and one leg 5 cm. Work out the other leg. (answer in cm)
  3. Grade 8 — An isosceles triangle has base 16 cm and two equal sides of 17 cm. Work out the perpendicular height of the triangle. (answer in cm)
  4. Grade 9 — A cuboid measures 3 cm × 4 cm × 12 cm. Work out the length of the longest straight rod that fits inside it (the space diagonal). (answer in cm)
  5. 9+ — A = (−3, 2) and B = (9, 7). Work out the exact distance AB.

Ask the person who made it

Inside Incandio you can put a question to Albert Einstein, who answers knowing exactly which module you have just finished: “As a boy you found your own proof of Pythagoras' theorem, and you called that experience a revelation about what thinking could do. What was it like to prove something eternal for yourself at twelve years old?”

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Pythagoras' Theorem, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Pythagoras' Theorem works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Pythagoras' Theorem against Euclid — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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