Pythagoras, the Pythagorean Theorem and Term Paper

It was then notated that if triangle ABC is a right triangle, with a right angle at C, then c2 = a2 + b2. Earlier, the converse of this theorem appears to have been used. This became proposition number 47 from Book I of Euclid's Elements ("Pythagorean," 2007).

Although this theorem is traditionally associated with Pythagoras, it is actually much older.

More than a millennium before the birth of Pythagoras, four Babylonian tablets were created demonstrating some knowledge of this theorem, circa 1900-1600 B.C.. At the very least, these works represent the knowledge of at least special integers known as Pythagorean triples that satisfy it.

In addition, the Rhind papyrus, created around 1650 B.C., shows that Egyptians had knowledge of the theorem as well. However, the first proof of this theorem is still credited to Pythagoras, despite the fact that some scholars believe it was independently discovered in several different cultures ("Pythagorean," 2007).

In Euclid's Book I of the Elements, the work ends with the famous 'windmill' proof of the Pythagorean theorem. In Book VI of the Elements, Euclid later gives an even easier demonstration of the theorem, "using the proposition that the areas of similar triangles are proportionate to the squares of their corresponding sides.Apparently, Euclid invented the windmill proof so that he could place the Pythagorean theorem as the capstone to Book I" ("Pythagorean," 2007).

The Pythagorean Theorem's Relation to the Area of Circles:

The Pythagorean theorem can be used to find the area of a circle. If a square is drawn within a circle, with the corners of the square touching the perimeter of the circle, and the length of the sides of the square are known, this information can be used to determine the area of the circle (See Figure 1). If the length of the sides of the square are x, then the hypotenuse of the right triangle, y, that can be formed connecting the corners of the square is: sqrt (2) * x, using the Pythagorean theorem.

A x2 + x2 = y2 Figure 1:

2x2 = y2 sqrt (2) * x

As y is also the diameter of the circle, it can be determined that the radius of the circle is:

radius = (sqrt (2) * x)/2

From this, the area of the circle can be determined, with the formula area = ?(radius2), so:

The area of the circle = (2?x2)/4

References

Meserve, B.E. (2007). Pythagoras, theorem of......