Find the missing side of a right triangle.
Enter two sides and see the third, the squares on every side, the angles, the area, and the classic a squared plus b squared equals c squared diagram, all live as you type.
The triangle
What this means
The Pythagorean diagram
Squares built on every side
Triangle scorecard
Pythagorean triples
Whole-number right triangles. Notice how a squared plus b squared always lands exactly on c squared.
| Side a | Side b | Hypotenuse c | a squared + b squared | c squared |
|---|
The theorem, explained
The Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, a² + b² = c², where c is the hypotenuse. This 2,500-year-old theorem, attributed to the Greek mathematician Pythagoras, is one of the most proven theorems in mathematics with over 400 known proofs.
How to Use the Theorem
To find the hypotenuse: c = √(a² + b²). To find a missing leg: a = √(c² - b²). Enter the two known sides and the calculator computes the third. For example, with sides 3 and 4: c = √(9 + 16) = √25 = 5.
Verifying Right Triangles
To check if a triangle is a right triangle, test whether a² + b² = c² (using the longest side as c). If equal, it is a right triangle. If a² + b² > c², it is acute. If a² + b² < c², it is obtuse. This calculator verifies automatically.
Beyond Two Dimensions
The Pythagorean theorem extends to 3D: d = √(a² + b² + c²) gives the space diagonal of a rectangular prism. It also generalizes to n dimensions. In its most general form, it underlies the distance formula and the concept of Euclidean distance.
Common questions
Does the Pythagorean theorem work for all triangles?
No, it only applies to right triangles (those with a 90° angle). For other triangles, use the law of cosines: c² = a² + b² - 2ab×cos(C), which generalizes the Pythagorean theorem.
What are some common Pythagorean triples?
The most common are (3,4,5), (5,12,13), (8,15,17), (7,24,25), and (9,40,41). Multiples of these also work: (6,8,10), (15,36,39), etc.
Who was Pythagoras?
Pythagoras was a Greek philosopher and mathematician (c. 570-495 BC). While the theorem bears his name, the relationship was known to Babylonians and Indians centuries earlier. His school formalized the proof.
How is the Pythagorean theorem used in real life?
Construction workers use it to ensure right angles (3-4-5 method). It is used in navigation (shortest distance), screen size measurement (diagonal), GPS calculations, and any situation requiring distance between two points.
The Pythagorean theorem applies only to right triangles. For any other triangle use the law of cosines. Results are rounded for display.