Pythagoras for children

Quick answer

Pythagoras was an ancient Greek mathematician who discovered a special pattern in right-angled triangles. If you have a right triangle, the square of the longest side (called the hypotenuse) equals the sum of the squares of the other two sides. This is called the Pythagorean theorem.

Learning outcome

By the end of this page, you will understand how to find the length of the longest side of a right triangle using the pattern discovered by Pythagoras.

• You can identify a right triangle by its square corner
• You can use the formula a² + b² = c² to find a missing side length
• You can solve simple problems involving right triangles in real life
• You can explain how this pattern works with small numbers

Why this matters

Pythagoras’s pattern helps us solve many problems in real life. Think about when you're building a fence, laying tiles, or flying a kite. You can use this pattern to make sure your corners are perfectly square and to measure distances accurately. It's used in construction, design, and even in space travel! Understanding this pattern helps you see how maths connects to the world around you.

How it works

Let's break down how Pythagoras’s pattern works:

1. First, we need a right triangle (a triangle with one square corner)
2. We label the two shorter sides as a and b
3. We label the longest side (the side opposite the square corner) as c
4. Then we use the formula: a² + b² = c²
5. We square each of the shorter sides (multiply them by themselves)
6. We add those two squared numbers together
7. The answer is the square of the longest side
8. To find the length of the longest side, we take the square root of that answer

Examples you can see

Let’s look at some simple examples:

These are called Pythagorean triples. Let’s check one:

• a = 3, so a² = 3 × 3 = 9
• b = 4, so b² = 4 × 4 = 16
• 9 + 16 = 25
• c² = 25, so c = √25 = 5

Try it yourself with the next example:

• a = 5, so a² = 5 × 5 = 25
• b = 12, so b² = 12 × 12 = 144
• 25 + 144 = 169
• c² = 169, so c = √169 = 13

Try it as a mini adventure

Adventure: The Lost Treasure Map

You and your friend are on a treasure hunt. You have a map that says:

• Start at the big oak tree
• Walk 3 steps east
• Walk 4 steps north
• Put a marker where you are
• Now walk back to the oak tree

How many steps do you need to walk back to the oak tree?

Draw this on paper:

• Draw a right triangle
• Label the eastward walk as 3 steps
• Label the northward walk as 4 steps
• Measure the straight line back to the tree

Using the pattern:

• a = 3, so a² = 9
• b = 4, so b² = 16
• 9 + 16 = 25
• √25 = 5

You need to walk 5 steps straight back to the tree!

Common mistakes

• Mistake: Forgetting to square the numbers before adding them
• Why it happens: Confusing multiplication with addition
• Fix: Always multiply a by a and b by b first, then add the results
• Mistake: Adding the numbers before squaring them
• Why it happens: Not understanding the order of operations
• Fix: Remember the formula is a² + b² = c², not (a + b)² = c²
• Mistake: Thinking that the longest side is always the one you add
• Why it happens: Not understanding what "square" means
• Fix: Remember that c² is the square of the longest side, not just the side that is longest

Parent and carer guide

Helping your child understand Pythagoras can be fun and easy:

• Start by drawing right triangles with your child and identifying the longest side
• Use simple examples like 3, 4, 5 to show the pattern
• Try real-life situations: measuring the diagonal of a TV, the length of a ladder against a wall
• Don’t worry if they make mistakes – they’re learning!
• Encourage them to draw triangles and label sides to see the pattern clearly
• Use everyday objects to show how the pattern works in real life
• Reassure them that it's okay to take time to understand this pattern
• Ask questions like: “What do you notice about the numbers?” or “Can you find another triangle that fits this pattern?”
• Remember: this is a pattern that works for all right triangles, not just the ones in the examples
• Don’t rush – let your child explore and play with the numbers

Practice questions

1. Find the missing side of a right triangle where a = 6, b = 8.
2. Find the missing side of a right triangle where a = 5, b = 12.
3. Find the missing side of a right triangle where a = 8, b = 15.
4. Find the missing side of a right triangle where a = 7, b = 24.
5. Find the missing side of a right triangle where a = 9, b = 40.

Answers:

1. c² = 6² + 8² = 36 + 64 = 100, so c = 10
2. c² = 5² + 12² = 25 + 144 = 169, so c = 13
3. c² = 8² + 15² = 64 + 225 = 289, so c = 17
4. c² = 7² + 24² = 49 + 576 = 625, so c = 25
5. c² = 9² + 40² = 81 + 1600 = 1681, so c = 41

FAQ

Who was Pythagoras?

Pythagoras was an ancient Greek mathematician who lived around 570–495 BC. He was interested in numbers and patterns, and he discovered that there was a special relationship in right triangles.

What is the Pythagorean theorem?

The Pythagorean theorem says that in a right triangle, the square of the longest side (called the hypotenuse) equals the sum of the squares of the other two sides. The formula is a² + b² = c².

Why do we need this?

We use this to measure distances, build structures, and solve problems in many real-world situations. It helps us make sure corners are square and to calculate lengths we can’t measure directly.

Is this only for triangles?

No, the pattern works for any right triangle, no matter how big or small. It's a universal rule that applies to all right triangles.

Can I use this in real life?

Yes! You can use it when measuring the diagonal of a TV, the length of a ladder, or even to help find the shortest path in a game or puzzle.