The pythagorean theorem is one of the most well-known geometric formulas. It states that in a right angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In other words, if you have a triangle with sides of length 3, 4 and 5, the pythagorean theorem says that 32 + 42 = 52.

The pythagorean theorem is named after the Greek mathematician Pythagoras, who is credited with discovering it. However, it is possible that the theorem was known long before Pythagoras. There are records of similar theorems in Babylonian and Chinese mathematics, and even in the Bible.

The pythagorean theorem has many practical applications. It is used in architecture and building, for example to calculate the length of beams or rafters. It is also used in navigation, to work out distances between two points using triangulation.

The pythagorean theorem can be written as a formula:

a^{2} + b^{2} = c^{2}

where a and b are the lengths of the shorter sides of the triangle, and c is the length of the hypotenuse.

The pythagorean theorem can also be used to find the area of a circle. The formula for the area of a circle is:

A = πr^{2}

where r is the radius of the circle.

The pythagorean theorem can be used to find the value of Ï€ (pi). This is because, if you take a circle with a diameter of 1 unit (meaning that the radius is 0.5 units), the pythagorean theorem tells us that the length of the circumference (the distance around the edge of the circle) is equal to Ï€.

We can also use the pythagorean theorem to find the slope of a line. The slope intercept form of a straight line is:

y = mx + c

where m is the slope of the line and c is the y-intercept (the point where the line crosses the y-axis).

The pythagorean theorem can be used to find the value of m. This is because, if we take a right angled triangle with one side on the x-axis and the other side on the y-axis, the pythagorean theorem tells us that the length of the hypotenuse is equal to the slope of the line.

- So, in summary, the pythagorean theorem is a very useful formula with many practical applications. It can be used to find the length of the hypotenuse of a right angled triangle, the area of a circle, the value of Ï€, and the slope of a line.