worldofcalcsPythagorean Theorem Calculator
Side c
5
Working
c² = a² + b² = 3² + 4² = 9 + 16 = 25
c = √25 = 5
3, 4, 5 is a Pythagorean triple: all three sides are whole numbers.
Area
6
Perimeter
12
What is the Pythagorean theorem?
In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle, and the longest side) equals the sum of the squares of the other two sides. If you know any two sides, you can find the third. It's used in geometry exams, construction, navigation, and anywhere you need a straight-line distance.
Pythagorean theorem
a² + b² = c² · c = √(a² + b²) · a = √(c² − b²)
a and b are the two shorter sides that meet at the right angle and c is the hypotenuse. To find the hypotenuse, add the squares; to find a shorter side, subtract the square of the known shorter side from the square of the hypotenuse.
Good to know
- The theorem only works for right-angled triangles. For other triangles, use the triangle solver, which applies the cosine rule.
- The converse is also true: if a² + b² = c², the triangle has a right angle. Builders use a 3-4-5 rope to mark a square corner.
- Multiplying a Pythagorean triple by any number gives another one: 3, 4, 5 becomes 6, 8, 10 and 9, 12, 15.
- The distance between two points (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²), the theorem applied on a graph.
- A square's diagonal is its side × √2, about 1.414 times the side.
Worked example: how high does a ladder reach?
- A 13 m ladder leans against a wall, with its foot 5 m from the wall
- The ladder is the hypotenuse: c = 13, and one side is b = 5
- a² = c² − b² = 13² − 5² = 169 − 25 = 144
- a = √144 = 12, so the ladder reaches 12 m up the wall
- Check: 5² + 12² = 25 + 144 = 169 = 13²
Common Pythagorean triples
Whole-number right triangles that come up often in exam questions.
| a | b | c | Check: a² + b² = c² |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 6 | 8 | 10 | 36 + 64 = 100 |
| 8 | 15 | 17 | 64 + 225 = 289 |
| 7 | 24 | 25 | 49 + 576 = 625 |
| 20 | 21 | 29 | 400 + 441 = 841 |
| 12 | 35 | 37 | 144 + 1225 = 1369 |
| 9 | 40 | 41 | 81 + 1600 = 1681 |
Multiples of these, such as 6, 8, 10, are triples too.