Pythagorean Theorem Calculator
Calculate the missing side of a right triangle using the Pythagorean theorem (a² + b² = c²). Perfect for students, engineers, and anyone working with right triangles in geometry, construction, or design.
Triangle Dimensions
Quick Examples:
Result
How to Use This Calculator
Understanding the Pythagorean Theorem
What is the Pythagorean Theorem?
The Pythagorean theorem is a fundamental principle in geometry that relates the three sides of a right triangle. It states that in a right triangle, the square of the hypotenuse (the longest side opposite the right angle) equals the sum of squares of the other two sides (called legs).
The Formula
- a and b are the lengths of the legs (shorter sides)
- c is the length of the hypotenuse (longest side)
Variations of the Formula
Applications
- Construction and carpentry (ensuring square corners)
- Navigation and GPS systems
- Computer graphics and game development
- Engineering and architecture
- Physics calculations involving vectors
- Distance calculations in coordinate geometry
Example Calculations
Example 1: Finding the Hypotenuse
A right triangle has legs of 3 feet and 4 feet. Find the hypotenuse:
Example 2: Finding a Leg
A right triangle has a hypotenuse of 13 meters and one leg of 5 meters. Find the other leg:
Example 3: Real-World Application
A ladder is leaning against a wall. The ladder is 10 feet long and the bottom is 6 feet from the wall. How high up the wall does the ladder reach?
Frequently Asked Questions
Does the Pythagorean theorem work for all triangles?
No, the Pythagorean theorem only applies to right triangles (triangles with a 90-degree angle). For other triangles, you would use the law of cosines or other geometric principles.
What are Pythagorean triples?
Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem. Common examples include (3,4,5), (5,12,13), (8,15,17), and (7,24,25). These are useful for quick calculations and checking your work.
How do I know which side is the hypotenuse?
The hypotenuse is always the longest side of a right triangle and is always opposite the right angle (90°). If you're looking at a right triangle, the hypotenuse is the side that doesn't touch the right angle corner.