Pythagorean Theorem Calculator
Find any side of a right triangle using the Pythagorean theorem a²+b²=c². Includes angles and area.
| a | b | c | Type |
|---|---|---|---|
| 3 | 4 | 5 | Primitive |
| 5 | 12 | 13 | Primitive |
| 8 | 15 | 17 | Primitive |
| 7 | 24 | 25 | Primitive |
| 6 | 8 | 10 | Multiple |
| 9 | 40 | 41 | Primitive |
| 11 | 60 | 61 | Primitive |
| 12 | 35 | 37 | Primitive |
| 20 | 21 | 29 | Primitive |
| 9 | 12 | 15 | Multiple |
Find the hypotenuse or a missing leg of any right triangle using a² + b² = c², complete with the resulting angles, area, and perimeter.
How Pythagorean Theorem Calculator Works
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². Enter the two legs to solve for the hypotenuse (c = √(a² + b²)), or switch modes to enter the hypotenuse and one leg to solve for the missing leg (b = √(c² − a²)).
Once all three sides are known, the calculator finds the two acute angles using inverse trigonometry — angle A = arcsin(a/c) — and since the third angle is fixed at 90°, angle B is simply 90° minus angle A.
It also reports the triangle's area (½ × a × b, since the legs are perpendicular) and perimeter (a + b + c), and shows a reference table of common Pythagorean triples — sets of whole numbers that satisfy the theorem exactly, like 3-4-5 and 5-12-13.
See It In Action
Who Uses Pythagorean Theorem Calculator and Why
- Verifying whether three measured side lengths actually form a right angle, such as squaring a foundation or deck corner during construction.
- Finding a missing wall, ladder, or diagonal length from two known measurements.
- Spot-checking a construction layout against a known Pythagorean triple like 3-4-5 or 5-12-13.
- Computing both acute angles once all three sides of a right triangle are already known.
Mistakes to Avoid
- Applying the theorem to a triangle that isn't actually a right triangle — a²+b²=c² only holds when one angle is exactly 90°.
- Mixing up which side is the hypotenuse when solving for a missing leg — only the hypotenuse (opposite the right angle) is the one being solved for as √(c² − a²) in that mode.
- Assuming a Pythagorean triple only counts in its exact form — any multiple of a known triple (like 6-8-10, doubled from 3-4-5) is also a valid triple.
Tips for Best Results
- Use the reference table of common triples (3-4-5, 5-12-13) to quickly spot-check a construction measurement without running the full calculation.
- Remember that only the hypotenuse gets isolated on one side of the equation — make sure you're solving for the right unknown side in 'Find a Leg' mode.
Fixing Common Problems
My result for a leg calculation looks off. — Confirm the value entered as the hypotenuse is actually the triangle's longest side and sits opposite the right angle — entering the wrong side as the hypotenuse throws off the whole calculation.
Terms Explained
Hypotenuse: The side opposite a right triangle's 90° angle, always the longest of the three sides.
Pythagorean triple: A set of three positive integers (a, b, c) that exactly satisfy a² + b² = c², like 3-4-5 or 5-12-13.