Solve right triangle side lengths (a² + b² = c²), acute angles, altitude, area, and perimeter with US & Metric unit conversions and exact radical support (k√r).
• Select your preferred US unit (inches, feet, yards, miles) or Metric unit (mm, cm, meters, km) in the numerical value section.
• Enter any 2 values (Leg a, Leg b, or Hypotenuse c) using whole numbers, decimals, or radical format (e.g. 2√5).
• Leave the third value blank to solve for it. The calculator derives all side lengths, altitude, area, perimeter, and equivalent conversions.
Select your unit and enter any two sides to solve the Pythagorean equation.
Select your unit and enter any two sides on the left (a, b, or c) to calculate the third side, angles, altitude, area, and perimeter.
Historical context, mathematical formulation, Pythagorean triples, and algebraic proofs.
Named after the ancient Greek mathematician Pythagoras (c. 570–495 BC), the Pythagorean Theorem states that in any right-angled triangle, the area of the square whose side is the hypotenuse (c) is equal to the sum of the areas of the squares on the other two legs (a and b):
A Pythagorean triple consists of three positive integers (a, b, c) that satisfy a² + b² = c². When the greatest common divisor GCD(a,b,c) = 1, it is a primitive Pythagorean triple.
Consider a large square with side length (a + b). The total area of this square can be computed in two ways:
No. The Pythagorean theorem applies exclusively to right-angled triangles (90°). For non-right triangles (acute or obtuse), the Law of Cosines (c² = a² + b² - 2ab·cos(C)) must be used instead.
In any Euclidean triangle, the longest side is always opposite the largest angle. Since the right angle (90°) is the largest possible angle in a right triangle, the hypotenuse (c) is strictly the longest side.
Using area equivalence: Area = ½ × a × b = ½ × c × h. Multiplying both sides by 2 and dividing by c yields h = (a × b) / c.
When the square root of a non-perfect square is factored into whole number roots (e.g. √20 = √(4 × 5) = 2√5), it is expressed in simplified radical form k√r.