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About the Pythagorean theorem calculator
In a right triangle (one angle is 90°), the two short sides are the legs a and b, and the long side across from the right angle is the hypotenuse c. The Pythagorean theorem says:
a² + b² = c²
So if you know two sides, you can always find the third. Choose what you want to find — the hypotenuse c, or one of the legs — and type the other two sides. You get the missing side with every step, the exact square root when there is one (√50 = 5√2), both other angles, the area, the perimeter and a drawing of your triangle.
Features
- Find any side — the hypotenuse c, or leg a, or leg b
- Steps with your numbers — 3² + 4² = 9 + 16 = 25, c = √25 = 5
- Exact roots — √50 is shown as 5√2 ≈ 7.0711
- Angles — α and β in degrees, worked out with the inverse tangent
- Area, perimeter and height — the height to the hypotenuse too
- Pythagorean triples — whole-number triangles like 3, 4, 5 are pointed out
- A drawing to scale — the triangle with its sides and angles
- Saved on your device — your triangle is still there next time
Worked examples
Find the hypotenuse. a = 3, b = 4.
c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5. The angles are α ≈ 36.87° and β ≈ 53.13°. The area is 3 × 4 ÷ 2 = 6.
An exact root. a = 5, b = 5.
c² = 25 + 25 = 50, so c = √50 = √(25 × 2) = 5√2 ≈ 7.0711. Both angles are 45°.
Find a leg: a ladder. A 13 ft ladder stands with its foot 5 ft from the wall. How high does it reach?
a² = 13² − 5² = 169 − 25 = 144, so a = √144 = 12 ft.
Corner to corner of a room. A room is 12 ft by 16 ft.
c² = 144 + 256 = 400, so the diagonal is √400 = 20 ft.
Pythagorean triples
These whole numbers make a right triangle. Any multiple works too (6, 8, 10 is 2 × 3, 4, 5).
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
| 9 | 40 | 41 |
Builders use 3-4-5 to check a square corner: mark 3 ft on one side and 4 ft on the other — if the diagonal is 5 ft, the corner is 90°.
Frequently asked questions
Which side is the hypotenuse?
The longest side, across from the right angle. It never touches the 90° corner. In a² + b² = c², it is c.
How do I find a leg instead of the hypotenuse?
Turn the formula around: a² = c² − b². Choose Find leg a (or b), then type the hypotenuse and the other leg.
Why do I get an error when I look for a leg?
The hypotenuse must be longer than each leg. If c is not the longest side, there is no right triangle with those sizes.
What does 5√2 mean?
It is the exact value of √50. The calculator takes the largest square out of the root: 50 = 25 × 2, and √25 = 5. As a decimal it is about 7.0711.
Does it work for triangles without a right angle?
No. a² + b² = c² is only true when one angle is exactly 90°.
How are the angles found?
α = tan⁻¹(a ÷ b), the angle opposite side a. The two small angles of a right triangle add up to 90°, so β = 90° − α.
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- Quadratic Formula Calculator — solve ax² + bx + c = 0 with steps
- Scientific Calculator — square roots, sin, cos and tan
How to use it
- Choose what to find: the hypotenuse c, leg a or leg b
- Type the other two sides
- Pick a unit if you want one
- Read the missing side in the result box, with the exact root, the angles and the area
- See every step and the drawing under the form
