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Pythagoras Theorem: Free Printable Worksheet for Year 9
- ForYear 9
- StageKey Stage 3
- CostFree
- FormatPrintable PDF
Pythagoras Theorem is a free printable maths worksheet for Year 9 (ages 13 to 14) in Key Stage 3. Students are given the two short sides of a right angled triangle and find the hypotenuse, the longest side. They use a squared plus b squared equals c squared, and practise squares and square roots along the way.
It comes from the Mini Minds Hub, the free library of printables from Jaques of London, the British games and toy company that has been making games for families since 1795. Print both pages, give page 1 to your student and keep page 2 for yourself. It has notes, help, a stretch idea and the answers. Black and white printing works well.
What is on the sheet
Each question on the sheet gives the two short sides of a right angled triangle, such as 3 and 4 or 7 and 24. Students sketch the triangle, label the hypotenuse, then square, add and take a square root to find the longest side. There is space to answer each question. The side lengths vary from one question to the next, so students meet a mix of small and larger numbers.
- Short sides 3 and 4
- Short sides 6 and 8
- Short sides 5 and 12
- Short sides 9 and 12
- Short sides 8 and 15
- Short sides 7 and 24
- Short sides 20 and 21
- Short sides 12 and 16
A worked example with different numbers
A right angled triangle has short sides of 30 and 40. Find the hypotenuse.
This is a different question from the ones on the sheet, with different numbers. It shows two ways to work.
Method 1: square, add, square root- Sketch the triangle. Mark the right angle and label the side opposite it c. The short sides are a = 30 and b = 40.
- Square each short side: 30 × 30 = 900 and 40 × 40 = 1600.
- Add the squares: 900 + 1600 = 2500. This is c squared.
- Find the square root of 2500. It is 50, because 50 × 50 = 2500.
- Draw a square on each side of the triangle.
- The two small squares have areas 900 and 1600.
- Their areas added together should equal the area of the square on the hypotenuse: 2500.
- The side of that square is 50, so the hypotenuse is 50.
Ways to build understanding
Sketch first
A quick drawing shows which side is opposite the right angle. That side is the hypotenuse and it is always c in the formula.
Think in areas
A squared means the area of a square drawn on that side. The two smaller squares together match the largest square. This gives the formula a meaning.
Say each step aloud
Square, add, square root. Saying the order helps students avoid skipping a step, especially the final square root.
Check the size
The hypotenuse must be longer than each short side, but shorter than the two short sides added together. This quick test catches many errors.
About the theorem
Pythagoras theorem links the three sides of a right angled triangle. The square of the longest side, the hypotenuse, equals the sum of the squares of the other two sides. It only works when one angle is exactly 90 degrees. It is named after the ancient Greek thinker Pythagoras, although people in other places knew about the relationship too.
The theorem is useful far beyond the classroom. Builders and surveyors use it to check that corners are square and to find straight line distances. It also underpins finding the distance between two points on a coordinate grid. In Key Stage 3, students move from remembering the formula to using it with confidence.
Did you know?
- The hypotenuse is always the longest side of a right angled triangle.
- The theorem is named after Pythagoras, an ancient Greek thinker.
- The theorem only works for triangles with a right angle.
- A square number is a number multiplied by itself.
- Builders and surveyors use the theorem to check corners and measure distances.
Skills and curriculum link
- Squares
- Square roots
- Right angles
- Using a formula
- Finding a missing length
- Checking answers
This sheet supports the area below. It is matched to the published England framework. Please check it against your own school plan.
| Framework | Subject or area | Strand | Year |
|---|---|---|---|
| National Curriculum (England) | Maths | Geometry and measures | Year 9 |
What children will be able to do
- I can name the longest side of a right angled triangle.
- I can use a squared plus b squared equals c squared.
- I can find a missing length.
Ideas that help with Pythagoras
- Spot the right angleThe hypotenuse sits opposite the right angle. It does not touch it. Finding the right angle first shows which side is which.
- Squares and roots are oppositesSquaring multiplies a number by itself. A square root undoes that. Knowing common squares by heart makes roots much quicker.
- Order mattersSquare both short sides, add them, then take the root. The root is the last step, not the first.
- Sense checkThe longest side must be longer than either short side. If an answer is shorter, something has gone wrong.
Key words
- HypotenuseThe longest side of a right angled triangle. It is opposite the right angle.
- Right angleAn angle of exactly 90 degrees, like the corner of a sheet of paper.
- Square numberThe result of multiplying a whole number by itself. 10 × 10 = 100.
- Square rootThe number that, multiplied by itself, gives the number you started with. The square root of 100 is 10.
- SquaredMultiplied by itself. a squared means a × a.
- LabelA letter or name written on a diagram to show which side or angle you mean.
How to use it
- Print both pages. Page 1 is for your student. Keep page 2, which has the notes and answers.
- Sketch each triangle before starting. Mark the right angle and label the hypotenuse, the side opposite it.
- Work through the first question together. Talk about each step: square, add, square root.
- Let your student try the rest alone, then check against page 2. Ask how they found each answer.
- Come back another day to redo any tricky questions. Repeating a familiar activity builds confidence.
Help, stretch and game ideas
If it feels too hard
Show the first example with a square drawn on each side of the triangle. Colour the squares in and talk about their areas. Then work out the squares, add them and find the root together. Let your student lead the next question.
To stretch confident children
Find the missing short side in a triangle where the hypotenuse and one short side are given. Ask what changes in the method. Then invent a triangle of your own, work out its sides and swap with a friend to check.
Make it a game
Hunt for right angled triangles around the house, such as a shelf bracket, a book corner or a folded tea towel. Measure the two short sides, predict the hypotenuse, then measure it to check your prediction.
Mistakes to watch for
Watch forA student adds the short sides and stops, or adds them without squaring.
Try thisRemind them that the formula uses squares. Square each side first, then add. Using 10 and 10 shows that adding gives 20, but squaring then adding gives 200.
Watch forA student forgets the final square root and writes the sum of the squares as the answer.
Try thisAsk whether the answer could be the length of a side. The sum is c squared, so it needs a square root to give c.
Watch forA student labels a short side as the hypotenuse.
Try thisGo back to the sketch. Trace the right angle with a finger, then point to the side directly opposite it.
Talk about it
- Which side is the hypotenuse, and how can you tell?
- Why do we square the sides?
- How did you find the square root?
- How can you check your answer makes sense for the triangle?
- Where in real life might someone need a missing length?
- What would you do differently if you knew the hypotenuse but not a short side?
Take it off the page
Right angle hunt
Use the corner of a sheet of paper to test for right angles around the home. List the objects that have them and sketch one.
Squares on squared paper
Draw a right angled triangle on squared paper and a square on each side. Count squares to compare areas.
Square number recall
Write the squares of the whole numbers from 1 to 15 on small cards. Mix them up and match each square to its root.
Who it is for
Parents
Sit with your child for the first question, then let them lead. Ask them to sketch the triangle before any calculating. Praise careful thinking, not just correct answers.
Teachers
Use it as a starter, plenary, homework or early finisher task. Page 2 has support and stretch ideas for mixed ability groups. Listen for reasoning when students explain which side is the hypotenuse.
Home educators
Print it as part of a wider maths plan. It pairs well with measuring tasks around the home and revision of squares and roots. Print again for brothers, sisters or revision later.
Where this fits
Before this, students work with square numbers, square roots, angles and right angles. They also meet area, which helps them picture a squared as a real square. Confident recall of squares makes this topic much smoother.
After this, students use the theorem to find a missing short side and to solve problems in context, such as distances and shapes. They also move on to trigonometric ratios in right angled triangles, which the Key Stage 3 programme of study includes as well.
Check it worked
Ask these when the sheet is finished. Listen for the reasoning, not just the answer.
- Which side is the longest, and how do you know?
- How did you find the missing side?
- How can you check the answer makes sense for the triangle?
Questions parents and teachers ask
- What age is Pythagoras Theorem for?
It is written for Year 9 (ages 13 to 14) in Key Stage 3. It suits students who know square numbers and square roots and are ready to apply them to right angled triangles.
- What will students practise?
Students use a squared plus b squared equals c squared to find the longest side of right angled triangles. Along the way they practise squares, square roots and spotting the right angle.
- How do I use it?
Print both pages and give page 1 to your student. Ask them to sketch each triangle and label the hypotenuse, the side opposite the right angle. Check the answers together using page 2.
- What if it feels too hard?
Draw a square on each side of the first triangle yourself, as page 2 suggests, and talk about their areas. Then work through the first question together. Let your student lead the next one.
- Is it free?
Yes. It is free to print for home and classroom use. You can print it again for brothers, sisters or revision a few weeks later.
- Can teachers use it in class?
Yes. It works as a starter, plenary, early finisher task, homework or table activity. It needs no preparation beyond printing, and page 2 has support and stretch ideas for mixed groups.
- Can I use it for home education?
Yes. Home educators and tutors can use it alone or as part of a wider maths plan. It works well alongside measuring tasks, and it can be printed again for revision.
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Made by Jaques of London, making games for families since 1795. A printable supports classroom learning and does not replace a teacher.