Pythagorean theorem
Calculate hypotenuse or leg.
Guide: Pythagorean Theorem
The Pythagorean theorem is one of the fundamental theorems of Euclidean geometry, describing the relationship between the sides of a right triangle.It states: in every right triangle, the sum of the squares of the legs equals the square of the hypotenuse. If we denote the legs as a and b, and the hypotenuse as c, then a² + b² = c². This theorem was already known in ancient Babylonian and Indian civilizations, but its name comes from the Greek mathematician Pythagoras.
Pythagorean triples are right triangles where all sides are integers.The simplest example is the triangle with sides 3, 4, and 5, where 3² + 4² = 5² (9 + 16 = 25). Other examples include 5, 12, 13 and 8, 15, 17. These triangles are particularly useful in practice because they can be constructed without needing precise measurements.
The Pythagorean theorem finds applications in many fields of science and engineering.In construction and architecture, it is used to determine right angles for foundations and walls. In GPS navigation, it calculates distances between points on a plane. In computer graphics, it determines distances between pixels and renders 3D scenes. In physics, it describes relationships between forces, velocities, and accelerations in circular motion.
To calculate the hypotenuse, we use the formula c = √(a² + b²).In the calculator, select the "Hypotenuse (c)" mode and enter the values of the legs. To calculate a leg, we use the formula a = √(c² - b²) or b = √(c² - a²). In the calculator, select the "Leg (a/b)" mode and enter the longer side (hypotenuse) and one known side.
It is important to remember that the theorem applies ONLY to right triangles.In other triangles, the relationship between the sides is different. In an equilateral triangle, the relationship is described by the law of cosines: c² = a² + b² - 2ab·cos(γ). The Pythagorean theorem is a special case of the law of cosines for a 90-degree angle (cos 90° = 0).