📏 Pythagorean Theorem Calculator
Calculate the missing side of a right triangle using the Pythagorean theorem. Enter any two sides to find the third side, plus area and perimeter.
The Pythagorean Theorem
The Pythagorean theorem is one of the most famous and useful theorems in mathematics, stating 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. Mathematically, this is expressed as a² + b² = c², where c is the hypotenuse and a and b are the other two sides.
This fundamental theorem, attributed to the ancient Greek mathematician Pythagoras, has countless practical applications in construction, navigation, engineering, and physics. Whether you're calculating distances, designing structures, or solving geometric problems, the Pythagorean theorem provides a reliable method for finding unknown measurements in right triangles.
Our calculator makes it easy to apply this theorem by automatically determining which side you need to calculate based on your inputs. Simply enter any two known sides, and the calculator will find the third side using the appropriate form of the Pythagorean theorem, along with additional triangle properties like area and perimeter.
Calculate Missing Side
Enter any two side lengths to calculate the third side:
Common Pythagorean Triples
Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem:
3-4-5 Triangle
3² + 4² = 5²
9 + 16 = 25 ✓
Most basic Pythagorean triple
5-12-13 Triangle
5² + 12² = 13²
25 + 144 = 169 ✓
Commonly used in construction
8-15-17 Triangle
8² + 15² = 17²
64 + 225 = 289 ✓
Another primitive triple
7-24-25 Triangle
7² + 24² = 25²
49 + 576 = 625 ✓
Useful for larger measurements
6-8-10 Triangle
6² + 8² = 10²
36 + 64 = 100 ✓
Multiple of 3-4-5 triangle
9-12-15 Triangle
9² + 12² = 15²
81 + 144 = 225 ✓
3× the basic 3-4-5 triple
Real-World Applications
🏗️ Construction
Builders use the 3-4-5 rule to ensure corners are perfectly square. By measuring 3 units on one side and 4 units on the other, the diagonal should be exactly 5 units for a perfect right angle.
📐 Architecture
Architects use the Pythagorean theorem to calculate roof slopes, staircase dimensions, and structural supports, ensuring buildings are safe and meet design specifications.
🧭 Navigation
GPS systems and navigation tools use the theorem to calculate the shortest distance between two points, helping determine travel routes and distances.
📺 Technology
Screen sizes are measured diagonally using the Pythagorean theorem. A screen that's 16 units wide and 9 units tall has a diagonal of √(16² + 9²) = √337 ≈ 18.4 units.
Frequently Asked Questions
What is the Pythagorean theorem used for?
The Pythagorean theorem is used to find unknown side lengths in right triangles. It's essential in construction for ensuring square corners, in navigation for calculating distances, in engineering for structural design, and in many other fields where precise measurements are crucial.
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, which is a generalization of the Pythagorean theorem.
What are Pythagorean triples?
Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem. The most famous is 3-4-5, but there are infinitely many such triples. They're useful in construction and engineering because they provide exact integer measurements for right triangles.