What is the Pythagorean Theorem Calculator?

The Pythagorean Theorem Calculator is a tool used to find the length of a missing side of a right-angled triangle. The theorem, famously attributed to the ancient Greek mathematician Pythagoras, describes a fundamental relationship between the three sides of a right triangle.

It states that the square of the hypotenuse (the side opposite the right angle, usually denoted as 'c') is equal to the sum of the squares of the other two sides (the legs, usually denoted as 'a' and 'b').

This calculator allows you to enter the lengths of any two sides of a right triangle, and it will calculate the length of the remaining side using the theorem. It's a vital tool in geometry, trigonometry, construction, navigation, physics, and many other fields where right triangles are encountered.

How Does This Calculator Work?

The calculator operates based on the Pythagorean theorem formula:

• Input: You need to provide the lengths of exactly two sides of the right triangle (a, b, or c). Leave the field for the side you want to calculate empty. Input values must be positive numbers.
• Validation: The calculator first checks that exactly two valid, positive numbers have been entered. It also performs a crucial check if the hypotenuse 'c' is one of the *inputs*: it ensures that 'c' is longer than the other provided side ('a' or 'b'). If not, it indicates an invalid right triangle setup, as the hypotenuse must be the longest side.
• Calculation: Depending on which side is missing, it uses the appropriate rearrangement of the formula:
  • If a and b are known: c = √(a² + b²)
  • If a and c are known: b = √(c² - a²)
  • If b and c are known: a = √(c² - b²)
• Display: The calculator shows the specific formula rearrangement used and the calculated length of the missing side.

Frequently Asked Questions (FAQs)

• Does this calculator only work for right-angled triangles?
  Yes. The Pythagorean theorem specifically applies *only* to triangles containing a 90-degree angle.
• How many side lengths do I need to enter?
  Exactly two positive side lengths. Leave one field blank.
• What happens if I enter all three sides?
  This calculator finds a *missing* side. Entering all three won't produce a result. Leave one field blank.
• What units should I use?
  Use any consistent unit (cm, inches, etc.) for both inputs. The result will be in the same unit.
• Why did I get an "Invalid Triangle Setup" error?
  This means you entered a value for the hypotenuse (c) that was not longer than one of the legs (a or b). The hypotenuse must be the longest side in a right triangle.
• Can I enter negative numbers?
  No, side lengths must be positive.
• Is this Pythagorean Theorem calculator free?
  Yes, this tool is free.