Pythagorean Theorem Solver - Calculate Triangle Sides
Calculate any side of a right triangle using the Pythagorean theorem (a² + b² = c²). Enter any two sides to find the third with instant step-by-step solutions.
Triangle Sides
Results
What is a Pythagorean Theorem Solver?
A Pythagorean Theorem Solver is a mathematical tool that calculates the sides of a right triangle using the famous formula a² + b² = c². This fundamental theorem in geometry allows you to find any missing side when you know the other two.
This calculator can solve for:
- Hypotenuse (c) - The longest side opposite the right angle
- Leg a - One of the two shorter sides forming the right angle
- Leg b - The other shorter side forming the right angle
For distance calculations and coordinate geometry, try our Distance, Midpoint & Slope Calculator for comprehensive analysis.
To work with circles and their properties, use our Circle Geometry Calculator for radius, diameter, and area calculations.
For linear equations and graphing, check our Slope & Intercept Calculator to find slope and y-intercept.
How the Pythagorean Theorem Works
The Pythagorean theorem formula is:
Where:
- a = length of one leg
- b = length of the other leg
- c = length of the hypotenuse
To solve for each side:
- c = √(a² + b²)
- a = √(c² - b²)
- b = √(c² - a²)
Key Concepts Explained
Right Triangle
A triangle with one 90-degree angle. The Pythagorean theorem only applies to right triangles.
Hypotenuse
The longest side of a right triangle, always opposite the right angle.
Legs
The two shorter sides that form the right angle in a right triangle.
Pythagorean Triples
Sets of three whole numbers that satisfy the theorem, like 3-4-5 or 5-12-13.
How to Use This Calculator
Enter Two Sides
Input any two known side lengths
Leave One Blank
The missing side will be calculated
Click Calculate
Get instant results with step-by-step solution
Benefits of Using This Calculator
- • Instant Calculations: Get accurate results for any right triangle in seconds.
- • Step-by-Step Solutions: Understand the calculation process with detailed explanations.
- • Multiple Solving Modes: Solve for any missing side - a, b, or c.
- • Error Validation: Ensures all inputs are valid and creates a proper right triangle.
Factors to Consider
1. Right Angle Requirement
The triangle must have a 90-degree angle. The theorem does not work for acute or obtuse triangles.
2. Positive Values Only
All side lengths must be positive numbers. Negative or zero values are not valid for triangle sides.
3. Triangle Inequality
The hypotenuse must be longer than either leg. If calculated values don't make sense, check your inputs.
Frequently Asked Questions (FAQ)
Q: What is the Pythagorean theorem?
A: The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (longest side) equals the sum of squares of the other two sides: a² + b² = c².
Q: When can you use the Pythagorean theorem?
A: The Pythagorean theorem only works for right triangles (triangles with one 90-degree angle). It cannot be used for triangles without a right angle.
Q: What is the hypotenuse?
A: The hypotenuse is the longest side of a right triangle, located opposite the right angle. In the formula a² + b² = c², the hypotenuse is represented by c.
Q: Can I solve for any side?
A: Yes, this calculator can solve for any missing side. Enter any two sides to calculate the third: c = √(a² + b²), a = √(c² - b²), or b = √(c² - a²).
Q: Does this work for non-right triangles?
A: No, the Pythagorean theorem only applies to right triangles. For non-right triangles, use the Law of Cosines or Law of Sines instead.