Pythagorean Theorem Calculator
Enter any two sides of a right triangle to find the third with a² + b² = c². You also get the exact square-root form, both acute angles, area, perimeter, the height to the hypotenuse, a scaled diagram and the full working, all calculated in your browser.
- Angle α (opposite a)
- —
- Angle β (opposite b)
- —
- Area
- —
- Perimeter
- —
- Height to hypotenuse
- —
- Hypotenuse segments
- —
Step-by-step
How to use the Pythagorean Theorem Calculator
- Choose what to solve: the hypotenuse c, leg a or leg b.
- Enter the two known sides; the missing side updates instantly.
- Pick units and decimal places if you need them.
- Press Solve triangle to confirm, then read the exact form, angles and step-by-step working.
- Copy the results or try one of the example triangles.
Pythagorean theorem formulas
| Find | Formula | Example |
|---|---|---|
| Hypotenuse c | c = √(a² + b²) | a = 5, b = 12 → c = √169 = 13 |
| Leg a | a = √(c² − b²) | c = 10, b = 6 → a = √64 = 8 |
| Leg b | b = √(c² − a²) | c = 4, a = 2 → b = √12 = 2√3 ≈ 3.4641 |
| Area | A = a × b ÷ 2 | a = 3, b = 4 → A = 6 |
| Height to hypotenuse | h = a × b ÷ c | 3-4-5 → h = 12 ÷ 5 = 2.4 |
| Angle α | α = atan(a ÷ b) | 3-4-5 → α ≈ 36.87°, β ≈ 53.13° |
Common Pythagorean triples
Whole-number right triangles are handy for exams and for checking square corners on site: measure 3 ft along one wall and 4 ft along the other, and the diagonal should be exactly 5 ft (or use 6-8-10 or 9-12-15 for larger layouts).
| a | b | c | Smallest angle |
|---|---|---|---|
| 3 | 4 | 5 | 36.87° |
| 5 | 12 | 13 | 22.62° |
| 8 | 15 | 17 | 28.07° |
| 7 | 24 | 25 | 16.26° |
| 20 | 21 | 29 | 43.60° |
| 9 | 40 | 41 | 12.68° |
Special right triangles
- 45°-45°-90°: the legs are equal and the hypotenuse is leg × √2 (≈ 1.4142). Legs of 1 give c = √2.
- 30°-60°-90°: the sides are in the ratio 1 : √3 : 2. The short leg is half the hypotenuse.
- Converse: if a² + b² = c² for three side lengths, the triangle is a right triangle. If a² + b² > c² it is acute, and if a² + b² < c² it is obtuse.
Need other roots? Use the Square Root Calculator, or the Scientific Calculator for trig functions.
Frequently asked questions
What is the Pythagorean theorem?
In a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the two legs: a² + b² = c². It only applies to triangles with a 90° angle.
How do I find the hypotenuse?
Square both legs, add them and take the square root: c = √(a² + b²). For legs 6 and 8: 36 + 64 = 100, so c = √100 = 10.
How do I find a missing leg?
Subtract the square of the known leg from the square of the hypotenuse, then take the square root: a = √(c² − b²). For c = 13 and b = 5: 169 − 25 = 144, so a = 12. The hypotenuse must be the longest side.
What is simplest radical form?
It writes a square root with the largest possible perfect square taken out. For legs 2 and 4, c = √20 = √(4 × 5) = 2√5 ≈ 4.4721. The calculator shows this exact form whenever the answer is not a whole number.
What are Pythagorean triples?
Sets of three whole numbers that satisfy a² + b² = c², such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Any multiple of a triple, like 6-8-10, is also a triple. Builders use 3-4-5 to check that corners are square.
How are the angles calculated?
With trigonometry: angle α opposite leg a is atan(a ÷ b), and the other acute angle is β = 90° − α. The three angles of the triangle always add up to 180°.