Pythagorean Theorem Calculator
Calculate the missing side of a right triangle using the Pythagorean theorem
Calculation Results
Missing Side Length
Pythagorean Formula
Triangle Type
Triangle Visualization
Geometry Tips
- The Pythagorean theorem only applies to right triangles
- The hypotenuse is always the longest side of a right triangle
- All side lengths must be positive numbers
- You can use this theorem to verify if a triangle is right-angled
- The theorem has applications in construction, navigation, and physics
Understanding the Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in geometry that describes the relationship between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
a² + b² = c²
This theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery, though evidence suggests it was known to Babylonian mathematicians over a thousand years earlier.
Applications of the Pythagorean Theorem
Construction
Ensuring corners are square in buildings and structures
Navigation
Calculating shortest distances between points
Technology
Used in computer graphics and game development
Surveying
Measuring distances in land surveying
How to Use the Pythagorean Theorem
Identify the Right Triangle
Ensure you’re working with a triangle that has a 90-degree angle
Label the Sides
Identify the hypotenuse (longest side, opposite the right angle) and the two legs
Apply the Formula
Use a² + b² = c², where c is the hypotenuse
Solve for the Unknown
Rearrange the formula as needed to find the missing side length
Real-World Examples
The Pythagorean theorem has numerous practical applications:
Distance Calculation
Finding the straight-line distance between two points on a map
Construction
Ensuring building foundations are square using the 3-4-5 triangle method
Navigation
Calculating the shortest route between two points
Entertainment
Used in computer graphics to calculate distances and angles
Frequently Asked Questions
No, the Pythagorean theorem only applies to right triangles (triangles with one 90-degree angle). For other types of triangles, different formulas like the Law of Cosines must be used.
The most common Pythagorean triple is 3-4-5, where 3² + 4² = 5² (9 + 16 = 25). Other common triples include 5-12-13, 6-8-10, and 7-24-25.
Yes, the theorem can be extended to three dimensions. The distance between two points (x₁,y₁,z₁) and (x₂,y₂,z₂) in 3D space is √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²].
This calculator provides precise results based on the mathematical formula. The accuracy depends on the precision of your input values. For most practical purposes, it’s perfectly accurate.
Side lengths cannot be negative. If you encounter a negative result, check your inputs. The calculator will alert you if the values don’t form a valid right triangle.