Use the relationship between the three sides of a right-angled triangle. Apply it to missing sides, diagonals and practical problems.
Foundation and Higher · Useful for students building fluency and applying the method in GCSE questions.
Core pathway. Core GCSE methods are useful across Foundation and Higher. Start with the core examples and questions.
Higher extension. More demanding selections are marked Higher extension: practice questions 2, 5, 7, 8. Use these after the core work, following your teacher’s guidance.
2 worked examples and 8 original practice questions · Allow 25–40 minutes · Read online or print. Higher extensions are labelled. No account or email needed.
Printing includes the method, examples and space for working, followed by a separate answer section.
A useful approach
Identify the right angle and the hypotenuse, the side opposite it.
Use a² + b² = c², with c as the hypotenuse. Subtract a square when finding a shorter side.
Take the positive square root and round only the final answer. Check that the hypotenuse is longest.
A mistake to watch for
Pythagoras applies to right-angled triangles. The answer to the squared calculation is not yet the side length.
Learn each method, then practise it
Read each line of working and explain why it follows from the previous line.
Find the hypotenuse
The hypotenuse is opposite the right angle. Add the squares of the shorter sides, then take the positive square root. For coordinate distances, first find the horizontal and vertical differences. To test whether a triangle is right-angled, check whether the two smaller squares add to the largest square. Use the calculated length in any requested perimeter or circumference.
Worked example 1
Pythagoras · Core GCSE skill · No calculator · 2 marks
Triangle ABC is a right-angled triangle with the right angle at B.
AB = 6 cm and BC = 8 cm.
Work out the length of AC.
Diagram for Pythagoras. The question states the required measurements and relationships.Read a text description of this diagram
Given diagram. Triangle ABC is a right-angled triangle with the right angle at B. AB = 6 cm and BC = 8 cm. Work out the length of AC. Use the labelled information; do not estimate measurements from the sketch.
AC is the hypotenuse, so use Pythagoras' theorem: AC2 = AB2 + BC2
Triangle DEF is a right-angled triangle with the right angle at E.
The hypotenuse DF = 20 cm and DE = 8 cm.
Work out the length of EF.
Give your answer correct to 1 decimal place.
Diagram for Pythagoras. The question states the required measurements and relationships.Read a text description of this diagram
Given diagram. Triangle DEF is a right-angled triangle with the right angle at E. The hypotenuse DF = 20 cm and DE = 8 cm. Work out the length of EF. Give your answer correct to 1 decimal place. Use the labelled information; do not estimate measurements from the sketch.
EF is a shorter side, so rearrange Pythagoras' theorem: EF2 = DF2 − DE2
Write your working on paper. Marks indicate how much working to show; these questions are self-marked and do not change saved practice results. Use squared paper for drawing questions.
Triangle PQR is a right-angled triangle with the right angle at Q.
PQ = 7 cm and QR = 9 cm.
Work out the length of PR.
Give your answer correct to 1 decimal place.
Diagram for Pythagoras. The question states the required measurements and relationships.Read a text description of this diagram
Given diagram. Triangle PQR is a right-angled triangle with the right angle at Q. PQ = 7 cm and QR = 9 cm. Work out the length of PR. Give your answer correct to 1 decimal place. Use the labelled information; do not estimate measurements from the sketch.
The ladder leans against a vertical wall, with the foot of the ladder on horizontal ground 2.5 m from the base of the wall.
Work out how far up the wall the top of the ladder reaches.
Diagram for Pythagoras. The question states the required measurements and relationships.Read a text description of this diagram
Given diagram. A ladder is 6.5 m long. The ladder leans against a vertical wall, with the foot of the ladder on horizontal ground 2.5 m from the base of the wall. Work out how far up the wall the top of the ladder reaches. Use the labelled information; do not estimate measurements from the sketch.
ABCD is a trapezium in which AB is parallel to DC.
Angle ADC = angle DAB = 90°.
AB = 9 cm, AD = 12 cm and DC = 14 cm.
Work out the perimeter of the trapezium.
Diagram for Pythagoras. The question states the required measurements and relationships.Read a text description of this diagram
Given diagram. ABCD is a trapezium in which AB is parallel to DC. Angle ADC = angle DAB = 90 degrees. AB = 9 cm, AD = 12 cm and DC = 14 cm. Work out the perimeter of the trapezium. Use the labelled information; do not estimate measurements from the sketch.
Pythagoras · Higher extension · Calculator allowed · 5 marks
A field ABCD is a trapezium in which AB is parallel to DC.
Angle DAB = angle ADC = 90°.
AB = 25 m, AD = 15 m and DC = 33 m.
Joe wants to put a fence around the whole perimeter of the field.
Fencing is sold in 12 m rolls. Each roll costs £19.50.
Work out the total cost of the rolls of fencing Joe must buy.
Diagram for Pythagoras. The question states the required measurements and relationships.Read a text description of this diagram
Given diagram. A field ABCD is a trapezium in which AB is parallel to DC. Angle DAB = angle ADC = 90 degrees. AB = 25 m, AD = 15 m and DC = 33 m. Joe wants to put a fence around the whole perimeter of the field. Fencing is sold in 12 m rolls. Each roll costs £19.50. Work out the total cost of the rolls of fencing Joe must buy. Use the labelled information; do not estimate measurements from the sketch.
Compare the reasoning as well as the final answer. Another correct method is valid. If a step is unclear, revisit an example before trying a similar question.
Workouts provide interactive practice; choose the tier your school has recommended.
Use this in class
Give students time to write the method, then compare their working with the answer section. Ask which step they would check first if their answer differs.
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