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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn8300Papers 1–3 · Higher tierGeometry and measuresPythagoras' theorem

Geometry and measures · Subtopic 10 of 17

Pythagoras' Theorem

In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides: $a^2+b^2=c^2$ . Add the squares to find the hypotenuse; subtract to find a shorter side.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 25 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Identify the hypotenuse as the side opposite the right angle.
Find the hypotenuse, or a shorter side, of a right-angled triangle.
Use Pythagoras in practical problems and to find the distance between two points.
Test whether a triangle is right-angled.

Pause and think

Draw squares on each side of a $3$ - $4$ - $5$ triangle. What are their areas, and what do you notice?

Section 2 of 14

Core idea

Start hereWill the ladder reach?

A 5 m ladder leans against a wall with its foot 1.5 m out. How high does it reach? The wall, ground and ladder make a right-angled triangle, and Pythagoras links its three sides.

The big idea

Right-angled triangle: $a^2+b^2=c^2$ , $c$ the hypotenuse (opposite the right angle).

1. The theorem

Squares on the sides of a 3-4-5 triangle

9+16=25

. The two smaller squares together have exactly the same area as the big one.

A proof by rearrangement

$\text{big square side }a+b$
Place four copies of the triangle inside.
$(a+b)^2=4\cdot\tfrac12ab+c^2$
Four triangles + tilted inner square of side $c$ .
$a^2+2ab+b^2=2ab+c^2$
Expand.
$a^2+b^2=c^2$
Cancel $2ab$ .

Worked exampleHypotenuse

Sides 5 and 12.

Goal: $c$ .
Strategy: Square, add, root.

$25+144=169$
Add squares.
$c=13$
Root.

Answer

13

✓ CheckLongest side ✓.

Worked exampleThe ladder

5 m ladder, foot 1.5 m out.

Goal: Height.
Strategy: Square, subtract, root.

$h^2=25-2.25=22.75$
Hypotenuse is the ladder.
$h=4.77\text{ m}$
Root.

Answer

4.77

✓ CheckLess than 5 ✓.

Worked exampleIs it right-angled?

Sides 7, 24, 25 and 6, 8, 11.

Goal: Yes/no.
Strategy: Converse: check $a^2+b^2=c^2$ .

$49+576=625=25^2$
Yes.
$36+64=100\neq121$
No (obtuse, since $121>100$ ).

AnswerFirst yes; second no.

✓ CheckLargest side always as

c

✓.

Common misconceptionAdding for a shorter side If the hypotenuse is known, subtract.

Pause and think

Distance between $(1,2)$ and $(7,10)$ ?

Summary

Square, add, root for hypotenuse.
Square, subtract, root for a shorter side.

Hypotenuse: Longest side, opposite the right angle.

HistoryOlder than Pythagoras. Pythagoras lived around

500

BC, but a Babylonian clay tablet known as Plimpton 322, written about

1800

BC, already lists sets of numbers that fit

a^2+b^2=c^2

. The theorem was also known in ancient India and China. Pythagoras' followers may have been the first to prove it in general.

Fun factBuilders' 3-4-5 trick. To mark a perfect right angle without a protractor, builders measure

3

m along one wall,

4

m along the other, and adjust until the diagonal is exactly

5

m. It works because

3^2+4^2=5^2

Section 3 of 14

Key formulae

Pythagoras' theorem

a^2+b^2=c^2

Distance between two points

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

TrickKnow the triples.

3,4,5

;

5,12,13

;

8,15,17

;

7,24,25

and their multiples (

6,8,10

;

9,12,15

) come up again and again. Spot one and you can write the answer down without a calculator.

Section 4 of 14

Worked examples

Example 1 · Ladder0 / 2 steps

5

m ladder leans against a vertical wall with its foot

1.5

m from the wall. How high up the wall does it reach?

Example 2 · Isosceles triangle0 / 2 steps

An isosceles triangle has equal sides of

10

cm and a base of

12

cm. Find its height.

Section 5 of 14

Explore the graph

A right-angled triangle

Loading the interactive graph…

Try this

The value below the sliders is the hypotenuse $\sqrt{a^2+b^2}$ .
Find all the whole-number pairs $a,b$ up to $12$ that give a whole-number hypotenuse.
Double both $a$ and $b$ . What happens to the hypotenuse?

Whole-number solutions to $a^2+b^2=c^2$ are called Pythagorean triples.

In the real worldScreen sizes are hypotenuses. A "

55

-inch" TV is

55

inches along the diagonal. With a

16:9

screen, Pythagoras gives a width of about

48

inches and a height of about

27

inches.

Section 6 of 14

Common mistakes

IncorrectAdding when finding a shorter side

CorrectSubtract

a^2=c^2-b^2

Incorrect

\sqrt{9^2+12^2}=9+12

Correct

\sqrt{225}=15

Square roots do not split over addition.

IncorrectTreating any side as

c

Correct

c

is opposite the right angle

It must be the longest side.

IncorrectForgetting the square root

Correct

c^2=225\Rightarrow c=15

225

c^2

, not

c

Adding the two longest sides

$a^2+b^2=c^2$ , where $c$ is opposite the right angle

Section 7 of 14

Exam tip

Exam tipLabel first. Mark the hypotenuse with a

c

before you write anything. Then decide: finding

c

? add. Finding another side? subtract. Write the line

c^2=a^2+b^2

with numbers in it to secure the method mark.

Section 8 of 14

Guided practice

SA-4024-TR-G012 marksL1 FoundationNon-calculatorHypotenuse

A right-angled triangle has shorter sides

9

cm and

12

cm. Find the length of the hypotenuse.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-4024-TR1-0012 marksL1 FoundationNon-calculatorHypotenuse

A right-angled triangle has shorter sides

6

cm and

8

cm. Find the hypotenuse.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TR1-0022 marksL1 FoundationNon-calculatorHypotenuse

A right-angled triangle has shorter sides

7

cm and

24

cm. Find the hypotenuse.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TR1-0042 marksL1 FoundationCalculatorHypotenuse

A right-angled triangle has shorter sides

5

cm and

9

cm. Find the hypotenuse, to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TR-0012 marksL1 FoundationNon-calculatorPythagoras problems

A right-angled triangle has hypotenuse

13

cm and one other side

5

cm. Find the third side.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TR1-0032 marksL1 FoundationNon-calculatorPythagoras problems

A right-angled triangle has hypotenuse

17

cm and one shorter side

8

cm. Find the third side.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TR1-0062 marksL1 FoundationNon-calculatorHypotenuse

Find the length of the diagonal of a rectangle

12

cm by

5

cm.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TR1-0052 marksL2 StandardCalculatorPythagoras problems

5

m ladder leans against a vertical wall. Its foot is

1.5

m from the wall on level ground. How high up the wall does it reach? Give your answer to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TR1-0072 marksL2 StandardNon-calculatorPythagoras problems

Find the distance between the points

(1,2)

and

(7,10)

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TR1-0093 marksL2 StandardNon-calculatorPythagoras problems

An isosceles triangle has two sides of

10

cm and a base of

12

cm. Find its perpendicular height.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-4024-TR-0113 marksL3 ExamCalculatorPythagoras problems

A ship sails

12

km on a bearing of

060^\circ

, then

8

km on a bearing of

150^\circ

. Find its distance from the starting point, to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$a^2+b^2=c^2$ , with $c$ the hypotenuse.
Hypotenuse: add the squares. Shorter side: subtract.
Distance between points: Pythagoras on the $x$ and $y$ differences.
If $a^2+b^2\ne c^2$ , the triangle is not right-angled.

Section 12 of 14

Quick quiz

Q1 · SA-4024-TR1-0081 markL2 StandardCalculatorPythagoras problems

Is a triangle with sides

9

cm,

12

cm and

16

cm right-angled?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: GCSE Mathematics 8300
Where: Geometry and measures
Expected: The same, on both tiers.
Calculator: Paper 1 non-calculator; Papers 2 and 3 calculator.
Typical style: Problem-solving questions often combine topics.
Open in: 8300 · Papers 1–3 8300 · Papers 1–3

Section 14 of 14

Past paper questions

1 question (4 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

4024/12 · Oct/Nov 2025Question 234 marks
Pythagoras in a square-based pyramid, giving a surd.
Question paper, page 20 Mark scheme

All 4024 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 8300, First teaching 2015 (official specification).