Pythagorean Theorem Calculator – Find Triangle Sides
Calculate sides of a right triangle using a² + b² = c²
How to Use
- Select which side you want to solve for
- Enter the two known side lengths
- Click calculate to find the missing side
- View the step-by-step calculation
What is the Pythagorean Theorem?
The Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The formula is expressed as: a² + b² = c², where c is the hypotenuse and a and b are the other two sides (legs) of the right triangle.
How to Use the Pythagorean Theorem
Depending on which side you need to find, you can rearrange the formula:
- To find the hypotenuse: c = √(a² + b²)
- To find side a: a = √(c² - b²)
- To find side b: b = √(c² - a²)
Practical Examples
Example 1: A right triangle has legs of 3 and 4 units. Find the hypotenuse.
c = √(3² + 4²) = √(9 + 16) = √25 = 5 units
Example 2: A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg.
a = √(13² - 5²) = √(169 - 25) = √144 = 12 units
Real-World Applications
- Construction and architecture for ensuring right angles
- Navigation and GPS calculations
- Computer graphics and game development
- Surveying and land measurement
- Physics problems involving vectors
Frequently Asked Questions
- What is the Pythagorean theorem formula?
- The Pythagorean theorem formula is a² + b² = c², where a and b are the two legs of a right triangle and c is the hypotenuse (the longest side, opposite the right angle).
- Can I use this calculator for non-right triangles?
- No, the Pythagorean theorem only applies to right triangles. For other triangles, you would need to use the Law of Cosines or Law of Sines.
- 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), and (8, 15, 17).
- Why must the hypotenuse be larger than the other sides?
- In a right triangle, the hypotenuse is always the longest side because it's opposite the largest angle (90°). If the hypotenuse were smaller, the triangle couldn't exist.