What is the Pythagorean Theorem Calculator?
The Pythagorean Theorem Calculator is an intuitive tool designed to find the unknown length of a right-angled triangle. Whether you know the lengths of the two legs (a and b) and need the hypotenuse (c), or you have the hypotenuse and one leg and need the other, this calculator instantly solves the equation for you.
A right triangle is any triangle containing a perfect 90° angle. In geometry, the Pythagorean theorem establishes a fundamental mathematical relationship between the three sides of a right triangle.
How to Use This Calculator
Using the calculator requires only two known variables out of the three total sides:
- Step 1: Identify the parts of your triangle. The legs are a and b, and the longest side opposite the right angle is the hypotenuse c.
- Step 2: Enter any two known values into their corresponding input fields. Leave the unknown third field entirely blank.
- Step 3: Click "Calculate Missing Side".
- Step 4: The tool will output the missing length alongside the triangle's total Area, Perimeter, and its non-right angles.
The Formula / The Science
The Pythagorean Theorem is named after the ancient Greek mathematician Pythagoras. It states that the area of the square formed by the longest side of the right triangle (the hypotenuse) is precisely equal to the sum of the areas of the squares formed by the other two sides.
Solving for c (Hypotenuse): c = √(a² + b²)
Solving for a (Leg): a = √(c² - b²)
Solving for b (Leg): b = √(c² - a²)
If you have a triangle with legs of 3 and 4, the math looks like this:
c = √(3² + 4²) = √(9 + 16) = √25 = 5
Frequently Asked Questions
No, the standard Pythagorean theorem (a² + b² = c²) only applies to right triangles (triangles that have one 90-degree angle). For non-right triangles, you must use a generalization of this theorem known as the Law of Cosines.
The hypotenuse is always the longest side of a right triangle. It is located directly opposite the 90-degree right angle. In the Pythagorean formula, the hypotenuse is represented by the variable c.
By mathematical law, the hypotenuse must always be the longest side of a right triangle. If you enter a leg value (a or b) that is greater than or equal to the hypotenuse (c), the calculator will return an error because such a triangle is physically impossible to construct.
Once all three sides are known, the area of a right triangle is simply half of the base multiplied by the height. Because the two legs (a and b) meet at a 90-degree angle, one serves as the base and the other as the height. The formula used is Area = 0.5 × a × b.
A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². The most famous and simple example is 3, 4, and 5 (3² + 4² = 5²). Other common triples include 5, 12, 13 and 8, 15, 17. Triangles with these side lengths will always be perfect right triangles.