pythagorean
A Pythagorean theorem calculator solves right triangles using the relation a² + b² = c², where c is the hypotenuse and a, b are the two legs — finding any one side when the other two are known. The ZTools Pythagorean Calculator handles all three modes (find c, find a, find b), accepts decimal inputs in any unit (meters, feet, inches), shows the worked equation step-by-step, and additionally computes the triangle's area, perimeter, and the two acute angles for a complete right-triangle solution.
Use cases
Construction and carpentry — verifying square corners
The 3-4-5 rule comes straight from Pythagoras. Use 3 ft along one wall, 4 ft along another, measure the diagonal — if it's exactly 5 ft, the corner is square. The calculator checks any dimensions.
TV and monitor diagonal sizing
A 16:9 TV with a 55" diagonal: width = 55 × 16/√(16²+9²) ≈ 47.94", height ≈ 26.96". Solve for any missing side when shopping or planning a wall mount.
Geometry and trigonometry homework
Students enter two sides, see the third instantly with the worked formula. Plus the angles, area, and perimeter — useful for full-problem checks.
Distance between two points
The distance formula d = √((x₂−x₁)² + (y₂−y₁)²) is Pythagoras applied to coordinates. Use the calculator with the two coordinate differences as the legs.
How it works
- Pick which side to find — "Find hypotenuse c", "Find leg a", or "Find leg b". The calculator hides the side it will compute and shows fields for the two known sides.
- Enter the known lengths — Decimals OK. Units don't matter as long as they're consistent — input cm and you'll get cm out.
- The formula is applied — For c: c = √(a² + b²). For a: a = √(c² − b²). For b: b = √(c² − a²). Negative results from invalid inputs (e.g., leg > hypotenuse) are flagged.
- Read the result and the work — Final answer plus the worked equation. Bonus: area = (a × b) ÷ 2, perimeter = a + b + c, angles = arctan(opposite ÷ adjacent).
Examples
Input: a = 3, b = 4, find c
Output: c = 5 (3² + 4² = 9 + 16 = 25, √25 = 5)
Input: a = 5, c = 13, find b
Output: b = 12 (13² − 5² = 169 − 25 = 144, √144 = 12)
Input: a = 6, b = 8, find c
Output: c = 10. Area: 24. Perimeter: 24. Angles: 36.87°, 53.13°, 90°.
Frequently asked questions
What is the Pythagorean theorem?
For any right triangle (one 90° angle), the square of the hypotenuse (longest side, opposite the right angle) equals the sum of squares of the two legs: a² + b² = c². It only applies to right triangles; for other triangles, use the law of cosines.
How do I find a leg if I know the hypotenuse and the other leg?
Rearrange to a² = c² − b², so a = √(c² − b²). Example: c = 13, b = 5 → a² = 169 − 25 = 144 → a = 12.
What's a Pythagorean triple?
Three positive integers (a, b, c) that satisfy a² + b² = c². Common examples: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Useful for verifying right angles in construction.
Does the theorem work for 3D space?
Yes — generalized to 3D as d² = x² + y² + z². To find the diagonal of a box with sides a, b, c: d = √(a² + b² + c²). Useful for shipping, packaging, room measurements.
How do I check if a triangle is a right triangle?
Identify the longest side as the candidate hypotenuse. Check whether longest² = other₁² + other₂². If yes, it's a right triangle; otherwise, it's acute (longest² < sum) or obtuse (longest² > sum).
Tips
- Always ensure inputs are in consistent units — mixing meters and feet gives nonsense.
- For very small or very large numbers, watch for floating-point precision; the calculator uses extended precision internally.
- The 3-4-5 trick (or any Pythagorean triple) is a fast way to lay out a square corner without a protractor.
- For non-right triangles, switch to the law of cosines: c² = a² + b² − 2ab·cos(C).
Try it now
The full pythagorean runs in your browser at https://ztools.zaions.com/pythagorean — no signup, no upload, no data leaves your device.
Last updated: 2026-05-05 · Author: Ahsan Mahmood · Edit this page on GitHub