Pythagoras: the squares are actual squares
What this lesson covers
- For right-angled triangles, Pythagoras’ theorem states that the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
- If (a,b,c) is a Pythagorean triple, then so is (ka,kb,kc), where k is a positive integer.
The most famous equation in school maths
- a squared plus b squared equals c squared.
- Almost everyone can say it. Fewer know what it means.
- It is not really an equation. It is a fact about area.
Build a square on each side
- Take a right-angled triangle with sides 3, 4 and 5.
- The square on the 3 holds 9. The square on the 4 holds 16.
- Nine and sixteen make twenty-five. And the big square is 25.
| Side | Square on it | Area |
|---|---|---|
| Side of 3 | square on it | 9 squares of area |
| Side of 4 | square on it | 16 squares of area |
| Together | 9 + 16 | 25 |
| Side of 5 | square on it | 25. The same. |
The two small squares fill the big one
- Exactly. Not approximately. For every right-angled triangle.
- That is the theorem, and it is a statement about area.
- The equation is just a short way of writing it down.
Naming the sides, so the equation reads
- The hypotenuse is the side opposite the right angle.
- It is always the longest one. That is worth holding on to.
- In a² + b² = c², c is always the hypotenuse.
| Part | Where it is | Note |
|---|---|---|
| The hypotenuse | opposite the right angle | always the longest |
| The two legs | the sides making the right angle | |
| The theorem | a² + b² = c² | c is the hypotenuse |
First use: is this triangle right-angled?
- Square the two shorter sides, add them.
- Square the longest. Do they match?
- Six, eight, ten: thirty-six plus sixty-four is a hundred. Yes.
| Sides | Check | Verdict |
|---|---|---|
| 6, 8, 10 | 36 + 64 = 100 | and 10² = 100. Yes |
| 5, 12, 13 | 25 + 144 = 169 | and 13² = 169. Yes |
| 8, 15, 17 | 64 + 225 = 289 | and 17² = 289. Yes |
Second use: finding the hypotenuse
- Legs of 9 and 12. Square both: 81 and 144.
- Add them: 225. That is c squared, not c.
- Square root it. Fifteen.
| Step | Working | Result |
|---|---|---|
| Legs 9 and 12 | 81 + 144 | 225 |
| So c² | 225 | |
| c | √225 | 15 |
It usually does not come out whole
- Legs of 5 and 7 give 25 plus 49, which is 74.
- Root 74 is about 8.6, and that is a perfectly good answer.
- Check it: bigger than 7, and not absurdly big. Sensible.
| Step | Working | Result |
|---|---|---|
| Legs 5 and 7 | 25 + 49 | 74 |
| c | √74 | about 8.60 |
| Sanity check | bigger than 7, smaller than 12 | reasonable |
Third use, and the one people get wrong
- Hypotenuse 13, one leg 5. Find the other leg.
- Add the squares and you get 13.9, which is longer than 13.
- A side longer than the hypotenuse is impossible. So subtract.
| Step | What happens | Value |
|---|---|---|
| Hypotenuse 13, one leg 5 | the wrong move | 169 + 25 = 194 |
| That gives | √194 ≈ 13.9 | longer than the hypotenuse |
| Impossible | so it cannot be adding | |
| The right move | 169 − 25 | 144 |
| The other leg | √144 | 12 |
Now, why is any of this true?
- Take a big square with sides a plus b.
- Fit four copies of the triangle inside, round the edges.
- The gap left in the middle is a square of side c.
| Step | What you do | Result |
|---|---|---|
| A big square | side a + b | |
| Put 4 triangles inside | around the edges | |
| The gap in the middle | a square of side c | area c² |
Now move the same four triangles
- Same big square. Same four triangles. Just rearranged.
- Push them into two corners instead of round the edges.
- Now the gaps are two squares: one of a, one of b.
| Step | What you do | Result |
|---|---|---|
| Same big square | side a + b | |
| Same 4 triangles | rearranged into two corners | |
| The gaps now | a square of a, and a square of b | area a² + b² |
| Same square, same triangles | so the gaps are equal | a² + b² = c² |
And that is the whole proof
- Same big square both times. Same four triangles both times.
- So the leftover space has to be the same both times.
- One time it was c². The other, a² + b². They are equal.
Some side lengths come out whole
- 3, 4, 5 works. So does anything you get by scaling it.
- Double it: 6, 8, 10. Triple it: 9, 12, 15. Both work.
- One triple gives you infinitely many.
| Triple | Scaled | Check |
|---|---|---|
| 3, 4, 5 | the original | 9 + 16 = 25 |
| × 2 | 6, 8, 10 | 36 + 64 = 100 |
| × 3 | 9, 12, 15 | 81 + 144 = 225 |
| × 10 | 30, 40, 50 | still works |
Worth recognising, but do not rely on them
- Three, four, five. Five, twelve, thirteen. Eight, fifteen, seventeen.
- Spotting one saves you a calculation.
- But most triangles are not triples. Know the method.
| Triple | Multiples | Note |
|---|---|---|
| 3, 4, 5 | and all its multiples | |
| 5, 12, 13 | 10, 24, 26 and so on | |
| 8, 15, 17 | ||
| Warning | most triangles are not triples | know the method too |
What it is actually for
- A five metre ladder, with its base three metres from the wall.
- How far up does it reach? The ladder is the hypotenuse.
- Twenty-five minus nine is sixteen. Four metres.
| Step | Working | Result |
|---|---|---|
| A 5 m ladder | base 3 m from the wall | |
| How high? | 25 − 9 | 16 |
| Height | √16 | 4 m |
What to take away
- Six things, and the fourth one is the classic lost mark.
| The theorem | is about areas of squares, not just an equation |
| c | is the hypotenuse, opposite the right angle, always longest |
| Finding the hypotenuse | add the two squares, then square root |
| Finding a shorter side | subtract, do not add |
| A sanity check | the hypotenuse must be the longest side |
| Scaling a triple | gives another triple, every time |
Your turn
- Pause here and try all four.
- Answers are in the description.
| Question | |
|---|---|
| 1 | Legs of 8 and 6. Find the hypotenuse. |
| 2 | The hypotenuse is 25 and one leg is 7. Find the other leg. |
| 3 | Is 9, 12, 15 a Pythagorean triple? |
| 4 | Scale the triple 5, 12, 13 by 4. |
Show answer
1. 10. Sixty-four plus thirty-six is a hundred, and the root of that is ten. 2. 24. This one is a subtraction, because the unknown is a shorter side. Six hundred and twenty-five minus forty-nine is five hundred and seventy-six, and its root is twenty-four. 3. Yes. Eighty-one plus a hundred and forty-four is two hundred and twenty-five, which is fifteen squared. It is 3, 4, 5 scaled by three. 4. 20, 48, 52.
The answers
- 64 plus 36 is 100.
- Question 2 subtracts, because it wants a short side.
- 9, 12, 15 is 3, 4, 5 scaled by three.
| Answer | |
|---|---|
| 1 | 10 |
| 2 | 24 |
| 3 | Yes |
| 4 | 20, 48, 52 |
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