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Pythagorean Theorem Calculator

Find the missing side of any right triangle. Calculate hypotenuse or leg, verify right triangles, and explore Pythagorean triples.

a² + b² = c²

Enter two legs to find the hypotenuse

Enter the hypotenuse and one leg to find the other leg

Enter all three sides to verify if they form a right triangle

Generate Pythagorean triples up to a maximum value

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Mastering the Pythagorean Theorem: Complete Guide

The Pythagorean theorem is a cornerstone of geometry, stating that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². This Professional Pythagorean Theorem Calculator helps you find missing sides, verify right triangles, and explore Pythagorean triples—all with clear step‑by‑step explanations and visual diagrams.

Finding the Hypotenuse

When both legs a and b are known, the hypotenuse c is c = √(a² + b²). For example, a=3, b=4 → c = √(9+16) = √25 = 5. The calculator also provides the triangle's area, perimeter, and acute angles.

💡 Finding a Missing Leg

If the hypotenuse c and one leg a are known, the other leg is b = √(c² − a²). Example: c=13, a=5 → b = √(169−25) = √144 = 12.

Verifying a Right Triangle

The converse of the Pythagorean theorem states: If three sides satisfy a² + b² = c² (with c being the longest), the triangle is right‑angled. The calculator checks this and also identifies acute (a²+b² > c²) or obtuse (a²+b² < c²) triangles.

Pythagorean Triples

A Pythagorean triple consists of three positive integers a, b, c that satisfy a² + b² = c². Primitive triples (like 3‑4‑5, 5‑12‑13) have no common factor. Multiples (6‑8‑10, 9‑12‑15) are also valid. Use the Triples tab to generate all primitive and scaled triples up to a chosen maximum c.

Special Right Triangles

  • 45‑45‑90 triangle: Legs are equal, hypotenuse = leg × √2.
  • 30‑60‑90 triangle: Sides are in the ratio 1 : √3 : 2.

Frequently Asked Questions

Can the Pythagorean theorem be used in 3D?

Yes, the 3D distance formula d = √(x² + y² + z²) extends the theorem. It applies twice: first in a plane, then to the third dimension.

How do I find the area of a right triangle?

Area = ½ × base × height. In a right triangle, the two legs serve as base and height, so Area = ½ × a × b.

What if my triangle doesn't satisfy a² + b² = c²?

Then it's not a right triangle. If a² + b² > c², it's acute (all angles < 90°); if a² + b² < c², it's obtuse (one angle > 90°).

How do I generate more Pythagorean triples?

Use the Triples tab. It uses Euclid's formula: for m > n > 0, a = m² − n², b = 2mn, c = m² + n². The calculator shows primitive and scaled triples.

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