Pythagoras theorem Quiz (Grade 4)

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Pythagoras theorem — Notes & Worked Examples Show / Hide

Explanation:

Pythagoras' theorem applies to any right-angled triangle where c is the hypotenuse (the longest side, always opposite the right angle): a² + b² = c².

Follow the 3-step routine: Square both given sides, Add them if finding the hypotenuse c (or Subtract them if finding a shorter side a or b), and finally take the Square Root.

Example 1: Find the hypotenuse x of a right-angled triangle with legs 6 cm and 8 cm.

Finding hypotenuse (add): x² = 6² + 8² = 36 + 64 = 100

Square root: x = √100 = 10 cm → Answer: 10 cm

Example 2: A right-angled triangle has hypotenuse 14 cm and one shorter side 9 cm. Find the other shorter side y to 1 d.p.

Solution:

Finding a shorter side (subtract):

y² = 14² − 9² = 196 − 81 = 115

Square root: y = √115 = 10.7238…

Answer: 10.7 cm

One Page Guide (PDF) Pythagoras theorem Worksheets

Question 1

[2 marks]

Calculate the length of x
Give your answer to 1 decimal place if needed

x = cm

Worked Solution & Answer

Pythagoras: a² + b² = c² (where c is the hypotenuse)
x = 17.5 cm

Answer: x = x = 17.5 cm cm

Question 2

[2 marks]

Calculate the length of y
Give your answer to 1 decimal place if needed

y = cm

Worked Solution & Answer

Pythagoras: a² + b² = c² (where c is the hypotenuse)
y = 9.4 cm

Answer: y = y = 9.4 cm cm

Question 3 AQA Jun 2017 2H Q15

[2 marks]

Sami is trying to work out the exact value of y using Pythagoras’ theorem.
Here is her working:
(2y)² = 6² + 8²
2y² = 36 + 64
2y² = 100
y² = 100 ÷ 2
y² = 50
y = √(50)

(a) What error has she made in her working? [1 mark]

(b) Kai works out that y = 5 Mel says,
“y cannot be 5 because the hypotenuse should be the longest side and the other sides are longer than 5 cm”
Is Mel correct? Tick a box
[ ] Yes [ ] No Give a reason for your answer. [1 mark]

Worked Solution & Answer

(a)

(2y)² = 4y²
She did not square the 2. B1

Answer (a): She wrote (2y)² as 2y² instead of 4y²

(b)

No, because the hypotenuse is 2y = 2 × 5 = 10 cm, which is longer than both 6 cm and 8 cm. C1

Answer (b): No

Question Paper (PDF) • Mark Scheme (PDF)

Question 4 IGCSE Nov 2021 1F Q14 / 1H Q2

[3 marks]

Here is a right-angled triangle.

Calculate the value of h.
Give your answer correct to 3 significant figures.

h =

Worked Solution & Answer

(Pythagoras’ theorem: a² + b² = c²)
h² = 52² – 30² = 2704 – 900 = 1804 M2
⇒ h = √1804 = 42.4735... = 42.5 (3 s.f.) A1

Answer: h = 42.5

Question Paper (PDF) • Mark Scheme (PDF)

Question 5 AQA Nov 2022 2F Q26

[3 marks]

Here is a triangle.

Use Pythagoras’ theorem to work out the value of y.
Give your answer as a decimal.

y = cm

Worked Solution & Answer

y² + 9² = 14² ⇒ y² + 81 = 196 M1
y² = 196 - 81 = 115 M1
y = √(115) ≈ 10.72 cm A1

Answer: y = 10.7 cm

Question Paper (PDF) • Mark Scheme (PDF)

Question 6 AQA Jun 2025 3F Q23

[4 marks]

Double right-angled triangles sharing common side. Right triangle has hypotenuse 13 cm, base 12 cm. Left triangle has base 9 cm, hypotenuse x cm. Show 10 < x < 11. [4 marks]

Use Pythagoras’ theorem to show that the value of x is between 10 and 11

Worked Solution & Answer

Right triangle has hypotenuse 13 cm and side 12 cm: M1
Shared vertical side h = √(13² – 12²) = √(169 – 144) = √25 = 5 cm M1
Left triangle has sides 9 cm and 5 cm with hypotenuse x:
x² = 9² + 5² = 81 + 25 = 106 ⇒ x = √106 M1
Since 10² = 100 and 11² = 121, and 100 < 106 < 121, we have 10 < √106 < 11
Thus x is between 10 and 11. C1

Answer: Shared side = 5 cm, x = √106 ≈ 10.3 cm, which is between 10 (√100) and 11 (√121)

Question Paper (PDF) • Mark Scheme (PDF)

Question 7 IGCSE Nov 2023 2F Q18 / 2H Q2

[4 marks]

The diagram shows triangle ABC and triangle ECD

ACD and EBC are straight lines.
AB = 10 cm AC = 8 cm EB = 5 cm CD = 14 cm ED = w cm
Work out the value of w
Give your answer correct to one decimal place.

w =

Worked Solution & Answer

(Pythagoras’ theorem: a² + b² = c²)
In right-angled triangle ABC: BC = √(10² − 8²) = √36 = 6 cm M2
Then EC = 5 + 6 = 11 cm M1
In right-angled triangle ECD: w = √(11² + 14²) = √(121 + 196) = √317 = 17.804... ≈ 17.8. A1

Answer: w = 17.8

Question Paper (PDF) • Mark Scheme (PDF)

Question 8 IGCSE Jun 2025 2F Q27 / 2H Q10

[5 marks]

ABD and ABC are right-angled triangles.

AB = 16 cm BC = 12 cm AD = 1.5 × AC
Find the length of CD
Give your answer correct to 3 significant figures.

Worked Solution & Answer

(Pythagoras’ theorem: a² + b² = c²)
AC = √(12² + 16²) = 20 cm M2
AD = 30 cm M1
BD = √(30² – 16²) = √644 ≈ 25.377 cm M1
CD = 25.377 – 12 = 13.4 cm A1

Answer: 13.4 cm

Question Paper (PDF) • Mark Scheme (PDF)

Question 9

[5 marks]

This rectangular frame is made from 5 straight pieces of metal.

The weight of the metal is 1.5 kg per metre.
Work out the total weight of the metal in the frame.

Worked Solution & Answer

(Pythagoras’ theorem: a² + b² = c²)
Diagonal = √(12² + 5²) = √(144 + 25) = √169 = 13 m M2
Total length = 12 + 12 + 5 + 5 + 13 = 47 m M2
Total weight = 47 × 1.5 = 70.5 kg A1

Answer: 70.5 kg

Question Paper (PDF) • Mark Scheme (PDF)

Question 10

[5 marks]

Here is a triangular prism.

Work out the volume of the prism.
Give your answer correct to 3 significant figures.

cm³

Worked Solution & Answer

(The volume of a prism is area of cross-section × length)
Base of triangle = √(8.4² – 7.2²) = √(70.56 – 51.84) = √18.72 = 4.32666 cm M2
Area of triangle = 0.5 × 4.32666 × 7.2 = 15.576 cm² M2
Volume = 15.576 × 18 = 280.37 cm³ → 280 cm³ (3 s.f.) A1

Answer: 280 cm³

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Score: 0 / 10 (0 / 35 marks)

Pythagoras theorem Worksheets Next: Angles in parallel lines →

Grade 4 Quizzes

24 Topics

4A - Compound interest	4B - Indices (index laws)
4C - Prime factors HCF and LCM	4D - Real life graphs
4E - Inequalities on a number line	4F - Solving inequalities
4G - Forming and solving equations	4H - nth term of a linear sequence
4J - Expanding and factorising	4K - Pythagoras theorem
4L - Angles in parallel lines	4M - Angles in polygons
4N - Surface area	4P - Volume of a cuboid
4Q - Volume of a prism	4R - Bearings
4S - Cylinders	4T - Plans and elevations
4U - Scatter graphs	4V - Probability
4W - Averages from tables	4X - Reverse mean
4Y - Area of a trapezium	4Z - Loci and construction