Plane Geometry - Bruce E. Shapiro

SECTION 42. AREA AND CIRCUMFERENCE OF CIRCLES 231Figure 42.2: Construction of inscribed regular 12-gon from regular hexagonby bisecting each central angle.Theorem 42.4 The sequence of perimeters L 1 , L 2 , ... converges to somenumber C.Proof. The sequence is bounded above and increasing. From a theorem ofcalculus, the sequence converges.Definition 42.5 Let Γ = C(O, r). The circumference of Γ is defined tobeC = limn→∞ L nTheorem 42.6 The value of the circumference only depends on the radiusand not the center point.Proof. Let Γ = C(O, r) and Γ ′ = C(O ′ , r).We can construct the perimeter of each sequence of inscribed 3 × 2 n -gonsH n and H n.′ Applying SAS to each sector, the corresponding triangles△P k OP k+1∼ = △P′kO ′ Pk+1′Since each triangle is congruent, the corresponding edges have equal length.Hence the L n = L ′ n for each n. Hence each sequence must converge to thesame limit.Theorem 42.7 Let r, r ′ > 0 be any positive numbers; and let C and C ′be the circumferences of concentric circles Γ = C(O, r) and Γ ′ = C(O, r ′ ).Then C/r = C ′ /r ′ .Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.