Mathematical Discoveries - The Value of Pi

The discovery of Pi using the exhaustion method:

While Archimedes is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics.

 

Archimedes used the method of exhaustion to approximate the value of pi.
 

While Archimedes is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Archimedes used the method of exhaustion to approximate the value of pi.

Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. Through proof by contradiction, he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answers lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of π.

He did this by drawing a larger regular hexagon outside a circle and a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step.

As the number of sides increases, it becomes a more accurate approximation of a circle. After four such steps, when the polygons had 96 sides each, he was able to determine that the value of π lay between 3¹⁄₇(approximately 3.1429) and 3¹⁰⁄₇₁ (approximately 3.1408), consistent with its actual value of approximately 3.1416. He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle (πr²).