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GCSE Maths / Edexcel

Pythagoras' theorem

Identify the hypotenuse, choose add or subtract, give rounded or exact answers, use the converse, and apply Pythagoras in 2D and Higher 3D contexts.

Geometry and MeasuresFoundation and HigherGrades 4 to 7Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Pythagoras' theorem

Pearson Edexcel GCSE Maths geometry G20: know, derive and apply Pythagoras' theorem in 2D; apply in 3D contexts for Higher.

Practise this topicUpload your answerAsk AI to explain thisTry exam-style questions
Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

Pythagoras' theorem only works in right-angled triangles. Before writing a formula, find the right angle first.

The hypotenuse is the longest side and is always directly opposite the right angle. It is the side the right-angle marker does not touch.

In a right-angled triangle, a² + b² = c², where c is the hypotenuse and a and b are the two shorter sides.

If you are finding the hypotenuse, add the two shorter side squares: c² = a² + b².

If you are finding a shorter side, subtract: missing side² = hypotenuse² - other shorter side².

After adding or subtracting the squares, square root to return to a length. A squared length such as 97 is not the final answer.

Most calculator answers are decimals. Keep the full calculator display until the last line, then round once to the accuracy requested.

If a question asks for exact form or surd form, do not use a decimal. Simplify the square root instead, for example √20 = 2√5.

The converse works backwards: if the two shorter side squares add to the longest side square, the triangle is right-angled.

Higher questions may hide the right triangle inside an isosceles triangle, a coordinate grid, a rectangle, or a cuboid.

Key ruleFind the hypotenuse by adding squares; find a shorter side by subtracting squares.

Diagram guide

abca²b²c²a² + b² = c²hypotenuse
What the theorem saysThe hypotenuse is opposite the right angle. The square on the hypotenuse has the same area as the two smaller squares together: a² + b² = c².
a5 cm13 cmshorter side: subtracta² = 13² - 5²
Add or subtract?Finding the hypotenuse means add the two shorter side squares. Finding a shorter side means subtract from the hypotenuse square.
35 cm12 cm37 cmNo right-angle marker: test the sides.12² + 35² = 37², so it is right-angled.
The converseWhen no right angle is marked, test the side lengths. Use the longest side as c and compare a² + b² with c².
9 cm9 cm15 cm15 cmhDrop the height to split the base in half.
Hidden right trianglesIn an isosceles triangle, a perpendicular height splits the base in half and creates two right-angled triangles.
A(1, 3)B(6, 11)58distance² = 5² + 8²
Distance between two pointsThe distance between two coordinates is the hypotenuse. Use the horizontal and vertical differences as the two shorter sides.
AG8 cm6 cm5 cmFind the base diagonal first, then the space diagonal.
Higher: 3D PythagorasFor a cuboid, find the base diagonal first, then use that diagonal with the height to find the space diagonal.

Worked examples

Find the hypotenuse

A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.

abca²b²c²a² + b² = c²hypotenuse
Example: hypotenuseThe missing side is opposite the right angle, so it is the hypotenuse. Add the two shorter side squares.
  1. c² = 6² + 8².
  2. c² = 36 + 64 = 100.
  3. c = √100 = 10.
  4. Finish with the answer and units.

Answer: 10 cm

Find a shorter side

A right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm. Find the other shorter side.

a5 cm13 cmshorter side: subtracta² = 13² - 5²
Example: shorter sideThe 13 cm side is the hypotenuse, so the missing shorter side is found by subtracting.
  1. a² = 13² - 5².
  2. a² = 169 - 25 = 144.
  3. a = √144 = 12.
  4. Check: the shorter side is less than the hypotenuse.

Answer: 12 cm

Decimal answer and rounding

Find the hypotenuse when the shorter sides are 9 cm and 4 cm. Give your answer to 2 decimal places.

abca²b²c²a² + b² = c²hypotenuse
Example: decimal answerKeep the unrounded square-root value until the final answer line.
  1. x² = 9² + 4².
  2. x² = 81 + 16 = 97.
  3. x = √97 = 9.84885...
  4. To 2 decimal places, x = 9.85 cm.

Answer: 9.85 cm to 2 decimal places

Exact answer in surd form

Shorter sides are 2 cm and 4 cm. Find the hypotenuse in its simplest exact form.

abca²b²c²a² + b² = c²hypotenuse
Example: exact formExact form means leave the answer as a simplified square root, not a rounded decimal.
  1. c² = 2² + 4².
  2. c² = 4 + 16 = 20.
  3. c = √20.
  4. √20 = √(4 x 5) = 2√5.

Answer: 2√5 cm

Use the converse

A triangle has sides 20 cm, 21 cm and 29 cm. Show that it is right-angled.

35 cm12 cm37 cmNo right-angle marker: test the sides.12² + 35² = 37², so it is right-angled.
Example: converse testNo right angle is marked, so test the side lengths. The longest side is 29 cm.
  1. Use the longest side as the hypotenuse candidate: 29 cm.
  2. 20² + 21² = 400 + 441 = 841.
  3. 29² = 841.
  4. The two values are equal, so the triangle is right-angled by the converse of Pythagoras' theorem.

Answer: Right-angled

Isosceles triangle area

An isosceles triangle has two equal sides of 13 cm and a base of 10 cm. Find its area.

9 cm9 cm15 cm15 cmhDrop the height to split the base in half.
Example: split the isosceles triangleDrop a perpendicular height. The base splits into two 5 cm halves and the height creates a right-angled triangle.
  1. Half the base is 10 ÷ 2 = 5 cm.
  2. In one right-angled triangle, h² = 13² - 5².
  3. h² = 169 - 25 = 144, so h = 12 cm.
  4. Area = 1/2 x 10 x 12 = 60 cm².

Answer: 60 cm²

Distance between two points

Find the distance between A(1, 2) and B(7, 10).

A(1, 3)B(6, 11)58distance² = 5² + 8²
Example: coordinate distanceThe horizontal change and vertical change form the two shorter sides of a right-angled triangle.
  1. Horizontal difference = 7 - 1 = 6.
  2. Vertical difference = 10 - 2 = 8.
  3. AB² = 6² + 8² = 36 + 64 = 100.
  4. AB = √100 = 10.

Answer: 10

Higher: 3D Pythagoras

A cuboid measures 3 cm by 4 cm by 12 cm. Find the length of the diagonal from one corner to the opposite corner.

AG8 cm6 cm5 cmFind the base diagonal first, then the space diagonal.
Example: two applications of PythagorasFirst find the base diagonal, then use it with the vertical height to find the space diagonal.
  1. Base diagonal² = 3² + 4² = 25, so the base diagonal is 5 cm.
  2. Space diagonal² = 5² + 12².
  3. Space diagonal² = 25 + 144 = 169.
  4. Space diagonal = 13 cm.

Answer: 13 cm

Common mistakes

  • Using Pythagoras on a triangle that is not right-angled.
  • Adding when finding a shorter side instead of subtracting.
  • Choosing the wrong side as the hypotenuse. It is opposite the right angle, not simply the side that looks slanted.
  • Forgetting to square root after adding or subtracting the squares.
  • Rounding too early instead of using the full calculator display until the final answer.
  • Missing units on the final answer line.
  • Giving a decimal when the question asks for exact form or surd form.
  • Stopping after the base diagonal in a 3D cuboid question.

Quick exercise

Try these before moving to the exam-style questions.

  1. Shorter sides are 9 cm and 12 cm. Find the hypotenuse.
    abca²b²c²a² + b² = c²hypotenuse
    Quick check: hypotenuseThe missing side is the hypotenuse, so add the two shorter side squares.
  2. Shorter sides are 5 cm and 7 cm. Find the hypotenuse to 2 decimal places.
    abca²b²c²a² + b² = c²hypotenuse
    Quick check: decimal hypotenuseWork out √(5² + 7²), then round once at the end.
  3. The hypotenuse is 25 cm and one shorter side is 7 cm. Find the other shorter side.
    a5 cm13 cmshorter side: subtracta² = 13² - 5²
    Quick check: shorter sideThe hypotenuse is known, so subtract the other side square.
  4. The hypotenuse is 11 cm and one shorter side is 6 cm. Find the other shorter side to 2 decimal places.
    a5 cm13 cmshorter side: subtracta² = 13² - 5²
    Quick check: decimal shorter sideUse √(11² - 6²), then round once at the end.
  5. A triangle has sides 9 cm, 12 cm and 15 cm. Is it right-angled? Type Yes or No.
    35 cm12 cm37 cmNo right-angle marker: test the sides.12² + 35² = 37², so it is right-angled.
    Quick check: converseCompare 9² + 12² with 15². If they are equal, the triangle is right-angled.
  6. Higher: shorter sides are 3 cm and 6 cm. Find the hypotenuse in simplest exact form.
    abca²b²c²a² + b² = c²hypotenuse
    Quick check: exact surd√45 simplifies because 45 = 9 x 5.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleFoundationCalculator3 marks

A ladder of length 6 m leans against a vertical wall. The foot of the ladder is 1.8 m from the base of the wall. How far up the wall does the ladder reach? Give your answer to 2 decimal places.

a5 cm13 cmshorter side: subtracta² = 13² - 5²
Question diagram: ladder against a wallThe ladder is the hypotenuse. The wall height is a shorter side, so subtract the ground distance square.
Pythagorasshorter sideladderrounding
Standard exam styleFoundation and HigherCalculator3 marks

A rectangle measures 11 cm by 6 cm. Find the length of its diagonal, correct to 3 significant figures.

abca²b²c²a² + b² = c²hypotenuse
Question diagram: rectangle diagonalThe rectangle diagonal is the hypotenuse of a right-angled triangle with shorter sides 11 cm and 6 cm.
Pythagorasrectangle diagonalsignificant figures
Standard exam styleFoundation and HigherCalculator3 marks

A triangle has sides of length 12 cm, 35 cm and 37 cm. Show that the triangle is right-angled.

35 cm12 cm37 cmNo right-angle marker: test the sides.12² + 35² = 37², so it is right-angled.
Question diagram: converse testNo right angle is marked. Use the longest side as the possible hypotenuse and compare the square totals.
Pythagorasconverseshow that
ChallengeHigherCalculator4 marks

An isosceles triangle has two sides of length 15 cm and a base of 18 cm. Work out the area of the triangle.

9 cm9 cm15 cm15 cmhDrop the height to split the base in half.
Question diagram: hidden heightDrop the perpendicular height. It splits the 18 cm base into two 9 cm halves.
Pythagorasisosceles triangleareahigher geometry
ChallengeHigherCalculator3 marks

Point A is (1, 3) and point B is (6, 11). Find the exact length of AB.

A(1, 3)B(6, 11)58distance² = 5² + 8²
Question diagram: coordinate distanceThe horizontal and vertical differences are the shorter sides of the right-angled triangle.
Pythagorascoordinate geometryexact formhigher
ChallengeHigherCalculator5 marks

A cuboid has AB = 8 cm, BC = 6 cm and CG = 5 cm. (a) Work out the exact length of the diagonal AG. (b) A straight pencil is 11 cm long. Will it fit completely inside the cuboid? Justify your answer.

AG8 cm6 cm5 cmFind the base diagonal first, then the space diagonal.
Question diagram: 3D PythagorasFind the base diagonal first, then use it with the height to find the longest internal diagonal.
Pythagoras3D geometrycuboidexact formhigher