The Pythagorean Theorem
In any right triangle, the squares of the two legs add up to the square of the hypotenuse: the side opposite the right angle. Know any two sides, and the third is forced.
1 The Theorem
Label the legs a and b, and the hypotenuse c. The theorem is one short equation — and it works in reverse, too.
Finding the hypotenuse: add the squares, then take the square root. Finding a leg: subtract instead — b² = c² − a² — then take the square root. The last step is always the square root.
Reverse (the converse): if three sides satisfy a² + b² = c², the triangle is right. That is how you test whether a triangle has a right angle.
Where it is used: ladder-against-wall problems, screen sizes (diagonal inches), navigation and maps, construction — and later, the distance formula. This is the most-used theorem in all of geometry.
2 See It
The areas tell the story — and ladder problems are right triangles in disguise.
3 Triples Worth Memorizing
A Pythagorean triple is three whole numbers that fit the theorem perfectly. Spotting one saves you the square root entirely.
| Triple | Check | Scaled cousins |
|---|---|---|
| 3 – 4 – 5 | 9 + 16 = 25 | 6–8–10, 9–12–15, 15–20–25 |
| 5 – 12 – 13 | 25 + 144 = 169 | 10–24–26 |
| 8 – 15 – 17 | 64 + 225 = 289 | — |
| 7 – 24 – 25 | 49 + 576 = 625 | — |
Scaling a triple by any whole number gives another triple: double 3–4–5 and you get 6–8–10, since 6² + 8² = 36 + 64 = 100 = 10².
4 Worked Examples
Follow each step. The pattern never changes: write the equation, isolate the square, take the root.
- Write the theorem: c² = 6² + 8².
- Add: c² = 36 + 64 = 100.
- Take the square root: c = √100 = 10.
- Check: 6-8-10 is the 3-4-5 triple scaled by 2. Correct.
- Write the theorem with c as the hypotenuse: 5² + b² = 13².
- Isolate: b² = 169 − 25 = 144.
- Take the square root: b = √144 = 12.
- Check: 5-12-13 is a classic triple. Correct.
- Write the theorem: c² = 2² + 7² = 4 + 49 = 53.
- 53 is not a perfect square — leave the exact form: c = √53.
- Check: (√53)² = 53 = 4 + 49. Exact beats decimal here.
- Model it: the ladder (9 ft) is the hypotenuse, the height h is a leg: 4² + h² = 9².
- Isolate: h² = 81 − 16 = 65.
- Take the square root: h = √65 ≈ 8.06 ft.
- Check: less than 9 ft (the ladder), more than 8 — sensible.
5 Common Mistakes
These three errors show up on almost every Pythagoras quiz.
c² = 100, so “the hypotenuse is 100.”
The work stopped one step early.
c² = 100, so c = √100 = 10.
The theorem gives you c² — the last step is always the square root.
The equation only holds when one angle is 90°.
No right angle, no Pythagoras. (That is what the Law of Cosines is for.)
5² + 13² = b² → b = √194.
The hypotenuse was treated as a leg.
5² + b² = 13² → b = √144 = 12.
c is always the hypotenuse — the longest side, opposite the right angle.
6 Quick Check
Try each one on paper first, then reveal the answer.
Key Points to Remember
- a² + b² = c², with c always the hypotenuse (longest side, opposite the right angle).
- Finding the hypotenuse: add the squares, then root. Finding a leg: subtract, then root.
- The last step is always the square root — never box c².
- Right triangles only. The converse tests whether a triangle is right.
- Memorize 3–4–5, 5–12–13, 8–15–17, 7–24–25 — and their scaled cousins.