The Pythagorean Theorem
The Pythagorean theorem is one of the most famous results in mathematics. It states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, written as a² + b² = c². This Pythagorean theorem calculator finds the missing side of a right triangle when you know the other two, and it also reports the two acute angles and the triangle's area.
How the Formula Works
The hypotenuse is always the longest side and sits across from the right angle. If you know the two legs, the hypotenuse is the square root of a² plus b². If you know the hypotenuse and one leg, the other leg is the square root of c² minus the known leg squared. The formula works in both directions, making it easy to solve for whichever side is missing.
Finding Angles
The Pythagorean theorem describes side lengths, but right triangles also have two acute angles that add up to 90 degrees. Using trigonometry, the angle opposite side a is found with the inverse sine of a divided by c. The other acute angle is simply 90 degrees minus the first. Together with the side lengths, these angles fully describe the triangle.
Real-World Applications
Builders use the Pythagorean theorem to check that corners are square, often with the 3-4-5 rule. Carpenters, roofers, and surveyors rely on it daily to make precise measurements without needing a protractor. It also forms the basis of the distance formula used throughout geometry and physics.
To measure the straight line between two coordinate points, try our distance calculator, which applies the same theorem in two and three dimensions.