OmniCalcsPythagorean Theorem Calculator
Find the hypotenuse or a missing leg of a right triangle with a² + b² = c².
How to Use This Pythagorean Theorem Calculator
Enter any two sides and leave the third blank. The calculator finds the missing side, then reports the area and perimeter of the right triangle.
- Legs a and b — the two sides that meet at the right angle.
- Hypotenuse c — the side opposite the right angle, always the longest.
- Finding c — c = √(a² + b²).
- Finding a leg — a = √(c² − b²), which requires c to be the longest side.
Practical Example
Legs of 3 and 4:
- c² = 3² + 4² = 9 + 16 = 25
- c = √25 = 5
- Area = ½ × 3 × 4 = 6
- Perimeter = 3 + 4 + 5 = 12
The 3-4-5 triangle is the classic Pythagorean triple, along with 5-12-13 and 8-15-17.
What Your Results Mean
- Hypotenuse — always the longest side; a leg can never be longer than it.
- Area — half the product of the legs, since they are perpendicular.
- Pythagorean triples — whole-number solutions like 3-4-5; their multiples (6-8-10) work too.
Frequently Asked Questions
What is the Pythagorean theorem?
For any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
How do I find a missing leg?
Rearrange the theorem: a = √(c² − b²). The hypotenuse must be longer than the known leg, otherwise no real triangle exists.
What is a Pythagorean triple?
A set of three whole numbers that fit the theorem exactly, such as 3-4-5, 5-12-13 or 8-15-17. Multiples of a triple (like 6-8-10) are also triples.
Can this work for any triangle?
No, only for right triangles. For other triangles use the law of cosines, which extends the theorem with an angle term.
Why must the hypotenuse be the longest side?
Because c² equals the sum of two positive squares, so c² is larger than either a² or b², making c the longest side.