Plane Geometry - Bruce E. Shapiro

SECTION 42. AREA AND CIRCUMFERENCE OF CIRCLES 233Sinceα(△P (n)kOQ (n)k) + α(△Q(n) k> α(△P (n)kOP (n)k+1 )(n)OPk+1 )(see figure 42.2) the sequence of areas is increasing.Since the sequence of areas is increasing an bounded (say, for example, bythe area of a circumscribed square), the sequence converges to a limit A.Definition 42.11 (Area of a Circle) Let Γ = C(O, r) be a circle. Thenα(Γ) = limn→∞ α(H n)Theorem 42.12 (Archimedes’ Theorem) Let Γ = C(O, r) be a circlewith circumference C. ThenA = 1 2 rCProof. Construct the sequence of inscribed hexagons as before, and definethe sequence of areas as A n .Since each H n is composed of n isosceles triangles of side length r andcentral angle 360/n, the area of each triangle is a n = b n h n /2 where b n isthe base of the triangle and h n is its height. HenceA n = n 2 b nh n = b nL n2where L n is the perimeter of H n . HenceA = limn→∞ A n= 1 2 lim b nL nn→∞= 1 ( ) ( )lim2b n lim L nn→∞ n→∞= 1 2 rCCorollary 42.13 Let Γ = C(O, r). Thenα(Γ) = πr 2Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.