The formula a2+b2 is commonly known as the Pythagorean Theorem. It is named after the ancient Greek mathematician Pythagoras, who discovered this theorem in the 6th century BC. The Pythagorean Theorem is one of the most famous and useful theorems in mathematics, and it has numerous applications in a variety of fields.

The Pythagorean Theorem states that in a right-angled 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. In other words, if we have a right-angled triangle with sides a, b, and c (where c is the hypotenuse), then:

## c2 = a2 + b2

This formula is represented visually in the famous Pythagorean Theorem diagram, which consists of a right-angled triangle with squares drawn on each of its sides. The area of the largest square (which is on the hypotenuse) is equal to the sum of the areas of the other two squares.

The Pythagorean Theorem has a wide range of applications in mathematics, physics, engineering, and many other fields. It is used extensively in geometry and trigonometry, and it is also used in calculus, where it plays a crucial role in the study of functions and derivatives.

One of the most common applications of the Pythagorean Theorem is in calculating distances. For example, if we know the lengths of two sides of a right-angled triangle (such as the horizontal and vertical distances between two points on a map), we can use the formula to calculate the distance between those two points. This is known as the distance formula.

The Pythagorean Theorem is also used in 3-dimensional geometry, where it is applied to right-angled triangles that are formed by the intersections of planes. It is also used in physics, particularly in mechanics, where it is used to calculate the displacement of an object moving in two dimensions.

The Pythagorean Theorem has many real-world applications as well. For example, architects and engineers use it to calculate the length of diagonal beams in buildings, and surveyors use it to measure the height of objects such as trees and buildings.

In conclusion, the formula a2+b2, or the Pythagorean Theorem, is a fundamental and incredibly useful tool in mathematics and many other fields. It has applications in geometry, trigonometry, calculus, physics, engineering, architecture, and more. By understanding the Pythagorean Theorem and its applications, we can solve a wide range of problems and gain a deeper appreciation for the beauty and power of mathematics.