Enter any two sides of a right triangle to calculate the missing side. You can also enter all three sides to verify whether they form a right triangle.
What Is the Pythagorean Theorem?
The Pythagorean theorem is one of the most important formulas in geometry. It applies to any right triangle, which is a triangle that has one 90° angle. The theorem states:
a2 + b2 = c2
Here, a and b are the two legs of the triangle, and c is the hypotenuse (the longest side, opposite the right angle).
How to Use This Pythagorean Theorem Calculator
Quick Steps
- Enter values for any two sides.
- Leave the side you want to solve for blank.
- Click Calculate.
- Read the missing side value, formula steps, and triangle details.
If you already know all three sides, enter all of them and the calculator will check if those sides satisfy the right triangle relationship.
Formulas for Each Case
1) Find the hypotenuse
If you know both legs: c = √(a2 + b2)
2) Find a missing leg
If you know the hypotenuse and one leg: a = √(c2 - b2) or b = √(c2 - a2)
Important: the hypotenuse must be larger than either leg, otherwise the triangle is not valid.
Worked Example
Suppose a triangle has legs of 6 and 8 units. To find the hypotenuse:
- Square each leg: 62 = 36, 82 = 64
- Add them: 36 + 64 = 100
- Take the square root: √100 = 10
So the hypotenuse is 10 units.
Real-World Uses
This theorem appears in far more places than math class. Common applications include:
- Construction (finding diagonal lengths, roof pitches, and framing distances)
- Engineering and architecture
- Computer graphics and game development
- Navigation, surveying, and mapping
- Physics problems involving vectors and displacement
Common Mistakes to Avoid
- Using the theorem on triangles that are not right triangles.
- Mixing units (for example, inches and feet) without converting first.
- Entering a hypotenuse value that is smaller than a leg.
- Forgetting to square values before adding or subtracting.
Tips for Accurate Results
Use consistent units
Keep all values in the same unit (meters, feet, centimeters, etc.). The result will be in that same unit.
Round thoughtfully
The calculator returns a rounded value for readability. For technical work, keep extra decimal places during intermediate steps.
Validate dimensions
If results seem off, double-check your input and ensure you are modeling a right triangle.
Frequently Asked Questions
Can this calculator solve any triangle?
No. It is specifically for right triangles. Non-right triangles require different methods such as the Law of Cosines.
What if I only know one side?
One side alone is not enough information. You need at least two sides, or one side plus another relationship (like an angle).
Can I use decimals?
Yes, decimals are fully supported.
Final Thoughts
A fast, reliable Pythagorean theorem calculator can save time and reduce errors, whether you're solving homework, planning a construction project, or writing technical software. Enter any two sides above and get instant results.