Home / IGCSE Maths 0580 / Trigonometry / Pythagoras' theorem
Cambridge IGCSE · Topic 6.1
IGCSE Maths 0580 / Topic 6 / IGCSE

Two squares make the third.

In every right-angled triangle the square on the longest side has exactly the same area as the squares on the other two sides put together. That one fact finds any missing side.

Full-screen labPythagoras and trigonometry lab → Topic 6All trigonometry lessons →
The key idea

In a right-angled triangle with hypotenuse \(c\) (the side opposite the right angle) and shorter sides \(a\) and \(b\): \(a^2 + b^2 = c^2\). To find the hypotenuse, add the squares; to find a shorter side, subtract.

16 cm² 9 cm² 25 cm² 4 cm 3 cm 5 cm 9 + 16 = 25
Fig. 1: a 3-4-5 triangle drawn to scale. The squares on the two shorter sides (9 cm² and 16 cm²) together equal the square on the hypotenuse (25 cm²)
Section 01

The square on the hypotenuse.

In the lab, drag the corners of the right-angled triangle and watch the three squares. Whatever shape you make, the two small areas always add up to the big one. Then make the angle not quite 90° and see the equation break.

Worked example Finding the hypotenuse
A right-angled triangle has shorter sides 7 cm and 11 cm. Find the length of the hypotenuse.
  1. The hypotenuse is the unknown, so add the squares: \(c^2 = 7^2 + 11^2\).
  2. \(c^2 = 49 + 121 = 170\).
  3. Square root: \(c = \sqrt{170} = 13.038\ldots\)
Answer: \(13.0\) cm (3 s.f.). Check: longer than 11 cm, as it must be.
Worked example Finding a shorter side
The hypotenuse of a right-angled triangle is 15 cm and one of the other sides is 9 cm. Find the third side.
  1. The unknown is a shorter side, so subtract: \(a^2 = 15^2 - 9^2\).
  2. \(a^2 = 225 - 81 = 144\).
  3. \(a = \sqrt{144} = 12\).
Answer: \(12\) cm
Worked example A ladder against a wall
A ladder 6.5 m long leans against a vertical wall. Its foot is on horizontal ground 1.8 m from the wall. How far up the wall does the ladder reach?
  1. The ladder is the hypotenuse (it is opposite the right angle between wall and ground).
  2. Height\(^2 = 6.5^2 - 1.8^2 = 42.25 - 3.24 = 39.01\).
  3. Height \(= \sqrt{39.01} = 6.2457\ldots\)
Answer: \(6.25\) m (3 s.f.)
Examiner trap

Adding the squares when you need a shorter side. If the hypotenuse is given, you must SUBTRACT. Using \(6.5^2 + 1.8^2\) for the ladder gives 6.74 m, longer than the ladder itself, which is impossible.

Section 02

Pythagoras everywhere.

Many shapes hide a right-angled triangle: the diagonal of a rectangle, the height of an isosceles triangle, the distance between two points on a grid. Sketch the triangle, mark the right angle, then use the theorem.

SituationThe hidden right-angled triangle
Diagonal of a rectangletwo sides of the rectangle and the diagonal
Height of an isosceles trianglethe height cuts the base in half
Distance between two pointshorizontal change, vertical change, and the line joining them
Is this triangle right-angled?test whether \(a^2 + b^2 = c^2\) with \(c\) the longest side (the converse)
Worked example Height and area of an isosceles triangle
An isosceles triangle has sides 13 cm, 13 cm and 10 cm. Find its height and its area.
  1. The height from the apex meets the base at its midpoint, making two right-angled triangles with base \(10 \div 2 = 5\) cm and hypotenuse 13 cm.
  2. Height\(^2 = 13^2 - 5^2 = 169 - 25 = 144\), so the height is 12 cm.
  3. Area \(= \tfrac{1}{2} \times 10 \times 12 = 60\).
Answer: Height 12 cm, area 60 cm2
Worked example Distance between two points
Find the distance between \(A(2, 1)\) and \(B(8, 9)\).
  1. Horizontal change: \(8 - 2 = 6\). Vertical change: \(9 - 1 = 8\).
  2. \(AB^2 = 6^2 + 8^2 = 36 + 64 = 100\).
  3. \(AB = \sqrt{100} = 10\).
Answer: \(10\) units
Worked example Is it right-angled?
Which of these triangles is right-angled? (a) 8 cm, 15 cm, 17 cm   (b) 7 cm, 9 cm, 12 cm
  1. (a) Longest side 17: \(8^2 + 15^2 = 64 + 225 = 289 = 17^2\). Equal, so right-angled.
  2. (b) Longest side 12: \(7^2 + 9^2 = 49 + 81 = 130\), but \(12^2 = 144\). Not equal, so not right-angled.
Answer: Only (a). In (b), \(130 < 144\), so the largest angle is more than 90°.
Your turn
A rectangle measures 9 cm by 4 cm. Find the length of its diagonal.
Examiner trap

For the isosceles triangle, use HALF the base. Using the full 10 cm gives \(\sqrt{169 - 100} = 8.31\) cm, which is wrong.

Stage 1 · Learn

Check your understanding

Four quick checks. Each correct answer earns XP and records this skill on your profile.

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In triangle \(PQR\), the right angle is at \(Q\). Which side is the hypotenuse?

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The shorter sides of a right-angled triangle are 5 cm and 12 cm. Find the hypotenuse.

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A right-angled triangle has hypotenuse 10 cm and one side 6 cm. Find the other side.

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Is a triangle with sides 6 cm, 7 cm and 9 cm right-angled?

Section 03

Exact answers and multi-step problems.

Without a calculator, leave the answer as a surd in its simplest form. In longer problems, keep full calculator values until the very end and round only the final answer.

Worked example Exact answer, without a calculator
Without a calculator, find the hypotenuse of a right-angled triangle with shorter sides 2 cm and 4 cm. Give your answer as a surd in its simplest form.
  1. \(c^2 = 2^2 + 4^2 = 4 + 16 = 20\).
  2. \(c = \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}\).
Answer: \(2\sqrt{5}\) cm
Worked example Taking the short cut
A rectangular field is 80 m long and 60 m wide. Sam walks along two edges from one corner to the opposite corner. Priya walks straight along the diagonal. How much further does Sam walk?
  1. Sam walks \(80 + 60 = 140\) m.
  2. Diagonal\(^2 = 80^2 + 60^2 = 6400 + 3600 = 10\,000\), so Priya walks \(\sqrt{10\,000} = 100\) m.
  3. Difference: \(140 - 100 = 40\) m.
Answer: \(40\) m further
Examiner trap

Rounding too early. If a first length is \(\sqrt{97} = 9.848\ldots\), keep it in the calculator for the next step; using 9.8 can change the final 3 s.f. answer.

Stage 2 · Exam

Exam-style questions

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A ladder of length 5 m leans against a vertical wall. The top of the ladder is 4.6 m above the horizontal ground. How far is the foot of the ladder from the wall?

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Find the distance between the points \(A(-3, 2)\) and \(B(5, -4)\).

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The diagonal of a rectangle is 20 cm and its width is 12 cm. Find the perimeter of the rectangle.

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An isosceles triangle has two sides of 10 cm and a base of 12 cm. Find its area.

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Go deeper · practice
Six original Cambridge-style questions
Hypotenuse and shorter sides, an exact surd answer, the converse, distance between points and a journey problem.