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The Pythagorean theorem as a puzzle

Use squares on triangle sides as a visual puzzle for older children.

Quick answer

The Pythagorean theorem says that in a right-angled triangle, if you square the two shorter sides and add them together, you get the square of the longest side. It's like a puzzle where the squares of the sides fit together perfectly.

Learning outcome

By the end of this page, you will understand how to use the Pythagorean theorem to find the missing side of a right-angled triangle, and you'll see how it's like solving a puzzle. Success signs:
  • You can identify the longest side of a right-angled triangle
  • You can use the formula a² + b² = c² to find a missing side
  • You can draw and label a right-angled triangle with its squares
  • You can solve simple puzzles using the theorem

Why this matters

The Pythagorean theorem is one of the most important ideas in maths. It helps builders make sure corners are square, helps sailors navigate, and helps architects design safe buildings. It's also a puzzle that you can solve with your own hands and eyes. Understanding it helps you see how maths works in the world around you.

How it works

The theorem says that if you have a right-angled triangle, the square of the two shorter sides (called 'a' and 'b') add up to the square of the longest side (called 'c'). Here's how to think about it:
  1. First, you need a right-angled triangle (a triangle with one 90-degree angle)
  2. Label the two shorter sides as 'a' and 'b'
  3. Label the longest side (the one opposite the right angle) as 'c'
  4. Then you can use the formula: a² + b² = c²
  5. If you know 'a' and 'b', you can find 'c' by adding a² + b² and then taking the square root
  6. If you know 'c' and one of the other sides, you can find the missing side

Examples you can see

Let's imagine you're building a treehouse. You need a ladder that reaches from the ground to the treehouse, which is 3 metres high. You place the ladder 4 metres away from the tree. How long does the ladder need to be? In this case:
  • One side of the right triangle is 3 metres (height)
  • The other side is 4 metres (distance from tree)
  • The ladder is the longest side (hypotenuse)
So:
  • a = 3
  • b = 4
  • c = ?
Using the formula:
  • a² + b² = c²
  • 3² + 4² = c²
  • 9 + 16 = c²
  • 25 = c²
  • c = √25 = 5
So the ladder needs to be 5 metres long. Let's try another one:
  • You have a right triangle where one side is 5 metres and the other is 12 metres
  • You want to find the longest side
Using the formula:
  • 5² + 12² = c²
  • 25 + 144 = c²
  • 169 = c²
  • c = √169 = 13
The longest side is 13 metres.

Try it as a mini adventure

Adventure: The Treasure Map Puzzle You're on a treasure hunt! You have a map that shows a right-angled triangle. The two shorter sides are marked as 6 units and 8 units. You need to find the distance to the treasure chest, which is at the longest point.
  1. Draw a right-angled triangle
  2. Label the two shorter sides as 6 and 8
  3. Label the longest side as 'x'
  4. Use the formula: 6² + 8² = x²
  5. Calculate: 36 + 64 = x²
  6. 100 = x²
  7. x = √100 = 10
The treasure is 10 units away from the starting point!

Common mistakes

  • Mistake: Forgetting to square the numbers before adding them
    Why it happens: Kids might add the numbers directly instead of multiplying them by themselves first
    Fix: Always square each number first, then add them together
  • Mistake: Confusing which side is 'c' (the longest side)
    Why it happens: Kids might not remember that 'c' is always opposite the right angle
    Fix: Always check that the longest side is marked as 'c'
  • Mistake: Not taking the square root correctly
    Why it happens: Kids might forget that if c² = 100, then c = 10, not 100
    Fix: Remember that to get back from a square to the original number, you take the square root

Parent and carer guide

This is a big idea for children, so take your time. Here's how you can help in 5-10 minutes:
  • Draw a right-angled triangle with your child and label the sides
  • Use blocks or drawing to show how squares fit on each side
  • Practice with simple numbers first (like 3, 4, 5)
  • Ask your child to explain what each letter means
  • Use real-life examples like the treehouse or treasure hunt
  • Don't worry if it takes a few tries - it's a new way of thinking
Remember: This theorem is not about memorizing a formula, but about seeing patterns and relationships. It's like a puzzle that helps you understand how shapes and numbers connect.

Practice questions

  1. Find the missing side of a right triangle where one side is 3 and the other is 4
  2. Find the missing side of a right triangle where one side is 5 and the other is 12
  3. Find the missing side of a right triangle where one side is 8 and the other is 15
  4. Find the missing side of a right triangle where one side is 7 and the other is 24
  5. Find the missing side of a right triangle where one side is 9 and the other is 40
Answers:
  1. 5
  2. 13
  3. 17
  4. 25
  5. 41

FAQ

What is the Pythagorean theorem?

The Pythagorean theorem says that in a right-angled triangle, the square of the two shorter sides equals the square of the longest side. It's written as a² + b² = c².

Who was Pythagoras?

Pythagoras was a Greek mathematician who lived around 570-495 BC. He was known for his work with numbers and shapes, and the theorem that bears his name was actually known to other cultures long before him.

Why do we need this theorem?

We use it in real life for building, navigation, surveying, and even video games. It helps us understand how distances and angles work in the world around us.

Can the theorem be used for any triangle?

No, it only works for right-angled triangles. The triangle must have one 90-degree angle for the theorem to apply.

How can I remember the formula?

A helpful way is to remember: "A squared plus B squared equals C squared." You can also think of it as: "The area of the squares on the two short sides equals the area of the square on the long side."