HMMT 二月 2000 · 团队赛 · 第 15 题

HMMT February 2000 — Team Round — Problem 15

解析

Consider a regular n -gon with radius r . Let x b e the side length of the n -gon. So, 2 sin e the en tral angle is (see diagram b elo w), use the La w of Cosines to nd that n q p 2 2 2 2 2 2 2 2 x = r + r 2 r r os , so x = 2 r (1 os ). Th us, x = r 2 1 os . n n n q p 2 So, the total p erimeter of the n -gon is nx = nr 2 1 os . No w, if w e tak e n lim of the p erimeter, the result will b e 2 n , sin e the n -gon approa hes a irle, n !1 q q p p 2 2 so lim nr 2 1 os = 2 r , and so lim nr 1 os = r 2 . n !1 n !1 n n x r r 2 π n