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Edexcel IGCSE Maths (4MA1) · Unit 5: Geometry

Edexcel IGCSE 4MA1 Pythagoras' Theorem: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

In a right-angled triangle $a^{2} + b^{2} = c^{2}$ , where $c$ is the hypotenuse: the longest side, opposite the right angle. To find the hypotenuse, square the two shorter sides, add and square root; to find a shorter side, subtract instead. The same idea gives the distance between two points and the longest diagonal of a cuboid.

Unit 5: Geometry
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Notes
Formulas
Worked examples
Mistakes
FAQ

Pythagoras' Theorem notes

Finding the hypotenuse

With shorter sides 9 cm and 40 cm: $c^{2} = 81 + 1600 = 1681$ , so $c = 41$ cm. The hypotenuse must come out as the longest side, which is a quick check.

Finding a shorter side

With hypotenuse 10 and one side 6: $b^{2} = 100 - 36 = 64$ , so $b = 8$ . Subtract whenever the hypotenuse is one of the sides you know.

Distance between two points

The distance between $(1, 2)$ and $(7, 10)$ is the hypotenuse of a right-angled triangle with sides 6 and 8: $\sqrt{6^{2} + 8^{2}} = 10$ . This is the length formula $\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$ .

Pythagoras in 3D

For a cuboid with sides $a$ , $b$ and $c$ , the space diagonal is $\sqrt{a^{2} + b^{2} + c^{2}}$ . A 2 cm by 3 cm by 6 cm cuboid has diagonal $\sqrt{4 + 9 + 36} = 7$ cm. In other 3D shapes, find a right-angled triangle inside the shape, often in two steps.

Is the triangle right-angled?

If the three sides satisfy $a^{2} + b^{2} = c^{2}$ , the triangle has a right angle. A triangle with sides 7, 24 and 25 is right-angled, because $49 + 576 = 625$ .

Key formulas

Pythagoras Learn it: $a^{2} + b^{2} = c^{2}$
Midpoint; length Learn it: $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}), \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$

Learn it = not printed on the 4MA1 Higher formulae sheet, so learn it by heart. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Pythagoras' Theorem worked examples

Worked example 1: Finding the hypotenuse

Find the hypotenuse of a right-angled triangle with shorter sides 8 cm and 15 cm.

Right-angled triangle with shorter sides 8 cm and 15 cm; the hypotenuse is unknown

Solution

$c^{2} = 8^{2} + 15^{2} = 289$ .
$c = 17$ cm.

3 steps

Tip: The hypotenuse is opposite the right angle.

Worked example 2: Ladder problem

A 6.5 m ladder leans against a wall with its foot 2.5 m from the wall. How high up the wall does it reach?

A 6.5 m ladder against a wall with its foot 2.5 m from the wall; the height h is unknown

Solution

The ladder is the hypotenuse: $h^{2} = {6.5}^{2} - {2.5}^{2} = 36$ .
$h = 6$ m.

3 steps

Tip: For a shorter side, subtract.

Worked example 3: Pythagoras in 3D

Find the length of the space diagonal of a 3 cm × 4 cm × 12 cm cuboid.

Cuboid 3 cm by 4 cm by 12 cm with a space diagonal drawn in red

Solution

$d^{2} = 3^{2} + 4^{2} + 12^{2} = 169$ .
$d = 13$ cm.

3 steps

Tip: Or find the base diagonal (5 cm) first.

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Pythagoras' Theorem exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 2 marks

A right-angled triangle has shorter sides of 9 cm and 12 cm. Work out the length of its longest side.

Show the answer and mark scheme

M1 9² + 12² (= 225)
A1 15 cm

Answer: 15 cm

Question 2 · easy · 2 marks

The longest side of a right-angled triangle is 25 cm and one of the shorter sides is 7 cm. Work out the length of the third side.

Show the answer and mark scheme

M1 25² − 7² (= 576)
A1 24 cm

Answer: 24 cm

Question 3 · medium · 3 marks

The longest side of a right-angled triangle is 17.5 cm and one of the other sides is 8.4 cm. Work out the length of the third side. Give your answer correct to 3 significant figures.

Show the answer and mark scheme

M1 17.5² − 8.4² (= 235.69)
M1 √235.69
A1 15.4 cm (15.35…)

Answer: 15.4 cm

Question 4 · medium · 3 marks

An isosceles triangle has two sides of 13 cm and a base of 10 cm. (a) Work out its perpendicular height. (b) Work out its area.

Show the answer and mark scheme

M1 13² − 5² (= 144)
A1 height 12 cm
B1 ½ × 10 × 12 = 60 cm²

Answer: (a) 12 cm (b) 60 cm²

Question 5 · medium · 3 marks

A rope 7.5 m long is tied to the top of a vertical pole and pegged to level ground 2.1 m from the foot of the pole. The rope is straight. Work out the height of the pole.

Show the answer and mark scheme

M1 7.5² − 2.1² (= 51.84)
M1 √51.84
A1 7.2 m

Answer: 7.2 m

Question 6 · medium · 3 marks

The screen of a television is a rectangle. Its diagonal is 140 cm and its width is 122 cm. Work out the height of the screen. Give your answer correct to 3 significant figures.

Show the answer and mark scheme

M1 140² − 122² (= 4716)
M1 √4716
A1 68.7 cm (68.67…)

Answer: 68.7 cm

Question 7 · medium · 3 marks

A boat sails 18 km due north and then 25 km due east. Work out its distance from its starting point. Give your answer correct to 3 significant figures.

Full worked solution Free, no sign-in

M1 18² + 25² (= 949)
M1 √949
A1 30.8 km (30.80…)

Answer: 30.8 km

Every question here has a full worked solution. See them all →

Question 8 · hard · 4 marks

A rectangle has a diagonal of length 30 cm. Its length is twice its width. Work out the area of the rectangle.

Show the answer and mark scheme

M1 x² + (2x)² = 30²
A1 x² = 180
M1 area = x × 2x = 2x²
A1 360 cm²

Answer: 360 cm²

Common Pythagoras' theorem mistakes

Adding when you should subtract. If the side you want is not the hypotenuse, subtract the squares.
Forgetting the square root. $c^{2} = 1681$ means $c = 41$ , not 1681.
Adding the sides before squaring. $9^{2} + 40^{2} = 1681$ , but $(9 + 40)^{2} = 2401$ . Square each side first.
No right angle. Pythagoras only works in a right-angled triangle. In any other triangle, use the cosine rule.

Where students lose marks across 4MA1, from the examiner reports →

Pythagoras' Theorem FAQ

Is Pythagoras' theorem on the 4MA1 formula sheet?

No. Pythagoras' theorem is not printed on the 4MA1 Higher formulae sheet, so learn it: a² + b² = c², with c the hypotenuse.

How do I use Pythagoras in 3D?

Find a right-angled triangle inside the shape. For the diagonal of a cuboid, find the diagonal of the base first and use it with the height, or use d² = a² + b² + c² in one step.

Can I use a calculator for Pythagoras' theorem in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Pythagoras' Theorem revision notes · 4MA1 formula sheet · Taking Cambridge 0580 instead? Cambridge 0580 pythagoras' theorem questions

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and every result was re-solved independently before publishing.

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Edexcel IGCSE 4MA1 Pythagoras' Theorem: exam questions and answers, notes and worked examples

Try it first

Pythagoras' Theorem notes

Finding the hypotenuse

Finding a shorter side

Distance between two points

Pythagoras in 3D

Is the triangle right-angled?

Key formulas

Pythagoras' Theorem worked examples

Worked example 1: Finding the hypotenuse

Solution

Worked example 2: Ladder problem

Solution

Worked example 3: Pythagoras in 3D

Solution

Pythagoras' Theorem exam-style questions

Question 1 · easy · 2 marks

Question 2 · easy · 2 marks

Question 3 · medium · 3 marks

Question 4 · medium · 3 marks

Question 5 · medium · 3 marks

Question 6 · medium · 3 marks

Question 7 · medium · 3 marks

Full worked solution Free, no sign-in

Question 8 · hard · 4 marks

Common Pythagoras' theorem mistakes

Pythagoras' Theorem FAQ

Next steps

Practise Pythagoras' theorem with your progress saved

More Unit 5 topics: Geometry