Pythagorean Theorem Calculator

The Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (c) — the side opposite the right angle — equals the sum of the squares of the other two sides (legs a and b): a² + b² = c². This is one of the most fundamental and widely used relationships in all of mathematics, with applications spanning construction, navigation, physics, computer graphics and engineering.

This free Pythagorean theorem calculator solves for any of the three sides — choose what to find, enter the two known values, and get the result with a complete 3-step solution. No signup required. For general triangle problems (non-right triangles), use our Triangle Calculator.

How to Use the Pythagorean Theorem Calculator Online

1. Choose what to find. Select Hypotenuse (c), Leg a, or Leg b from the mode tabs.
2. Enter the two known values. Labels update automatically based on your selected mode. Enter any positive numbers — decimals are supported.
3. Read the result. The missing side appears instantly in the results panel along with the triangle area.
4. Follow the steps. The 3-step solution shows the formula, arithmetic and final answer.
5. Check common triples. The reference table lists six classic Pythagorean triples for verification or practice.

Finding Each Side — Three Formulas

• Hypotenuse: c = √(a² + b²) — square both legs, add, take the square root.
• Leg a: a = √(c² − b²) — square the hypotenuse, subtract leg b squared, take the square root. Requires c > b.
• Leg b: b = √(c² − a²) — square the hypotenuse, subtract leg a squared, take the square root. Requires c > a.

How the Calculator Works Technically

The solver uses JavaScript's Math.sqrt() for square roots and standard exponentiation (v ** 2) for squaring. For leg calculations, it validates that the hypotenuse is larger than the known leg before computing to avoid negative square roots. Results are formatted to 4 decimal places. The step-by-step panel shows the intermediate squared values so you can verify each arithmetic step manually.

Pythagorean Triples and Their Properties

Primitive and Non-Primitive Triples

A Pythagorean triple (a, b, c) satisfies a² + b² = c² with all three values being positive integers. Primitive triples (GCD = 1) are generated by: a = m²−n², b = 2mn, c = m²+n² for integers m > n > 0 where m and n are coprime and not both odd. For m=2, n=1: (3, 4, 5). For m=3, n=2: (5, 12, 13). Every non-primitive triple is a multiple of a primitive one: (6, 8, 10) = 2 × (3, 4, 5). Use our LCM & GCD Calculator to verify that GCD(3,4,5) = 1.

6 Real-World Applications of the Pythagorean Theorem

• Construction and carpentry: The 3-4-5 rule is used to verify right angles on building sites — if the diagonal of a 3m × 4m rectangle is exactly 5m, the corner is square.
• Navigation: Straight-line distance between two points on a grid: d = √(Δx² + Δy²). GPS systems use 3D extensions of this formula.
• Computer graphics: Pixel distance, collision detection, and vector magnitude all use the Pythagorean formula in 2D and 3D.
• Physics: Resultant force from perpendicular components: F = √(Fx² + Fy²). Vector addition in any physics problem reduces to the Pythagorean theorem.
• Engineering: Diagonal bracing in structures uses right triangle geometry to compute brace length. Slope and grade calculations in road design use the same formula.
• Astronomy: Parallax distance measurements and stellar coordinate transformations both rely on right-triangle geometry. Pair with our Scientific Calculator for trig-based variants.

Privacy and Security

This Pythagorean theorem calculator runs entirely in your browser. No data is transmitted to ToollyX servers at any point.

✓Verified by ToollyX Team · Last updated June 2026