How the value of pi can be calculated to 20 places of decimal?
π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, due to Archimedes, is to calculate the perimeter, Pn , of a regular polygon with n sides circumscribed around a circle with diameter d. Then i.e., the more sides the polygon has, the closer the approximation. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. π can also be calculated using purely mathematical methods. Most formulas used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry and calculus. However, some are quite simple, such as this form of the Gregory-Leibniz series: . While that series is easy to write and calculate, it is not immediately obvious why
