The area of a circle of radius 1 is a transcendental number (that is, a number that cannot be obtained by the process of solving algebraic equations) denoted by the Greek letter π. To calculate its value, we may use a limiting process in which π is the limit of a sequence of known numbers. The method used by Archimedes was to inscribe in the circle a sequence of regular polygons. As the number of sides increased, so the polygon ‘filled’ the circle. Show, by use of the trigonometric identity cos 2θ = 1 – 2 sin²θ, that the area a_n of an inscribed regular polygon of n sides satisfies the equation

Show that a₄ = 2 and use the recurrence relation to find a₆₄.

Transcribed Image Text:

2 ()--[-] = 1- n 2 n (n > 4)