Pythagoras – GCSE Maths

Foundation and Higher · Grade 4

Jump to the questions and answers · Download the worksheet (PDF)

In any right-angled triangle the two shorter sides (the ones that touch the right angle) and the hypotenuse (the side opposite the right angle) are locked together by $a^2+b^2=c^2$. Square both shorter sides and add to get the hypotenuse squared. If you already know the hypotenuse, square it and subtract the other known square to get the missing shorter side.

The method is the same whether the triangle is drawn as a ladder against a wall, a diagonal of a rectangle, or two points on a coordinate grid. Sketch, mark the right angle, decide whether you are adding squares or subtracting them, then root at the end. Forgetting the square root is the classic way to throw the accuracy mark after doing the hard part.

Exact answers such as $\sqrt{65}$ are often wanted on the non-calculator paper. On a calculator paper, round only at the last line to 1 decimal place or 3 significant figures as asked, and write the units.

Objectives

• a² + b² = c², where c is the hypotenuse.
• Square both legs, add, then square-root.
• Subtract the known square from the hypotenuse squared, then square-root.

Key points

• Identify the hypotenuse — longest side, opposite the right-angle square.
• Find the hypotenuse with $c=\sqrt{a^2+b^2}$ (square, add, root).
• Find a shorter side with $a=\sqrt{c^2-b^2}$ (square, subtract, root).
• Use Pythagoras on a coordinate grid, a rectangle diagonal, or a ladder against a wall.

Worked examples

Questions and answers

GCSE Maths pythagoras questions and answers. Try each one, then open the answer. Easy questions come first.

Common mistakes

Most lost marks on this topic are one slip.

1. Adding the squares when you were finding a shorter side.

   Subtract from the hypotenuse squared.

2. Square-rooting before you add.

   Add the squares first, then take one square root.

3. Using Pythagoras on a triangle with no right angle.

   The theorem needs a right angle.

Exam tips

• This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
• If the question says “show that”, the result is given. The marks are for the steps that reach it.
• The method is the same on AQA-style and Edexcel-style papers.
• The first method mark is almost always the squared equation written out: $c^2=6.4^2+8.7^2$. Write it even on a calculator paper.
• Keep the full square root on the calculator until the final rounding. Rounding $10.800$ to $10.8$ too early can look like 1 d.p. when 2 d.p. was asked.

Printable worksheet

The same questions are in a PDF you can print. Answers start on a later page.

Download worksheet (PDF)

Or use Print in your browser. Answers under each question are included when you print.

Questions students ask

Related topics

• Angles in parallel lines
  Grade 4 · Foundation and Higher
• Angles in polygons
  Grade 4 · Foundation and Higher
• Bearings
  Grade 4 · Foundation and Higher
• Cylinders
  Grade 4 · Foundation and Higher
• Loci and constructions
  Grade 4 · Foundation and Higher

More on Pythagoras

• Pythagoras worked examples and method
  The method in order, key facts and the full lesson explanation
• Pythagoras practice questions
  A second set of questions with answers, plus quick checks

AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.