Right Triangle Calculator

The Formula

The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. The formula can be rearranged to solve for any side: c = √(a² + b²) when solving for the hypotenuse, or a = √(c² − b²) when solving for a leg. Combined with SOH CAH TOA trigonometry, you can also find all angles of any right triangle from just two side lengths.

Variable Definitions

How to Use This Calculator

1. 1

   Select what you want to find — hypotenuse, leg a, leg b, or the acute angles.

2. 2

   Enter the known values for the other two sides or angle.

3. 3

   View the missing side, calculated area, perimeter, and both acute angles.

4. 4

   The SOH CAH TOA trigonometry steps show exactly how each angle is derived from the side ratios.

5. 5

   The SVG right triangle dynamically scales based on the side lengths, showing labeled sides and angles.

The Pythagorean theorem relates the three sides of any right triangle.

Understanding the Concept

The Pythagorean theorem is one of the most fundamental relationships in Euclidean geometry, with a history dating back over 3,500 years to Babylonian mathematics. It applies exclusively to right triangles (triangles with one 90° angle). The theorem states that the area of the square on the hypotenuse equals the sum of the areas of the squares on the two legs — a² + b² = c². When solving for a leg, the formula is rearranged to a = √(c² − b²). The angles are found using SOH CAH TOA trigonometry: sin(α) = opposite/hypotenuse, cos(α) = adjacent/hypotenuse, tan(α) = opposite/adjacent. The theorem has practical applications in construction (squaring corners), navigation (shortest distances), physics (vector magnitudes, projectile motion), and computer graphics (distance calculations, collision detection). The Pythagorean theorem is so fundamental that it appears in nearly every branch of mathematics and science.

Frequently Asked Questions

Sources & References