Solve right triangles using the Pythagorean theorem: a² + b² = c².
Last updated: April 2026 | By Patchworkr Team
Enter any two values to find the third
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
This fundamental relationship has been used for over 2,000 years and applies to all right triangles.
Input any two of the three triangle sides (a, b, or c). You can leave one field empty. Stay with consistent units (meters, feet, etc.).
Why: The calculator needs at least two known values to solve for the third using the relationship a² + b² = c². Mixing units would invalidate the result.
Determine which side is missing: Are you finding the hypotenuse (c) from legs (a and b)? Or finding a leg from the hypotenuse and the other leg? This guides your calculation approach.
Why: Finding c requires addition under a square root (√(a² + b²)), while finding a leg requires subtraction (√(c² - b²)). The formula changes based on what you're solving for.
Press the Calculate button to apply the Pythagorean theorem. The calculator will perform the necessary arithmetic and display the result instantly.
Why: Computation errors are common when squaring and taking square roots manually. The calculator eliminates math mistakes and provides precise decimal accuracy.
Check the formula shown in the result area. It displays the exact calculation performed (e.g., √(3² + 4²) = √25 = 5) so you can verify the logic and learn the process.
Why: Understanding the formula step-by-step reinforces the theorem's logic and helps you detect input errors if a result seems wrong.
Use the result for construction layout, diagonal verification, navigation calculations, or geometry proof verification. Compare your result to expected values to ensure it's reasonable.
Why: Converting math results to real-world applications ensures your work solves actual problems. Sanity-checking results (the hypotenuse should be larger than either leg) catches logical errors.
A construction crew is framing a roof and needs to verify that the corner they've created is exactly 90 degrees. They measure the horizontal run as 4 meters and the vertical rise as 3 meters. They calculate the diagonal to verify alignment before continuing.
The crew has measured the two legs of the right triangle and needs to find what the hypotenuse (diagonal) should be.
Begin the Pythagorean theorem by squaring the first leg (the horizontal distance).
Square the second leg (the vertical distance).
Add the two squared values to get the square of the hypotenuse.
Take the square root of the sum to find the hypotenuse length: √(25) = 5 meters exactly.
Verify: 3² + 4² = 9 + 16 = 25 = 5² ✓ The equation balances perfectly, confirming this is a valid 3-4-5 Pythagorean triple.
The diagonal should measure exactly 5 meters. The crew now uses a 5-meter measuring tape or laser to check the actual diagonal. If it measures 5.0 m, the corner is truly 90°. If it measures 4.9 or 5.1 m, the corner is slightly off and needs frame adjustment. This simple verification technique ensures structural accuracy without needing a protractor and is commonly used throughout construction.
Sets of three integers that satisfy a² + b² = c². Examples: (3,4,5), (5,12,13), (8,15,17).
No, the Pythagorean theorem only applies to right triangles (with a 90° angle).
Yes, the theorem works with any real numbers, not just integers.
It's been known for at least 2,500 years, though named after Pythagoras.
If a² + b² = c², then the triangle is a right triangle. This helps verify right triangles.
No. By definition, c is the longest side in a right triangle.
Over 300 mathematical proofs exist for the Pythagorean theorem!
Yes: construction, navigation, surveying, computer graphics, and engineering rely on it.
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