HMMT 二月 2014 · 冲刺赛 · 第 3 题

英文解析

3. [ 4 ] Let ABCDEF be a regular hexagon. Let P be the circle inscribed in 4 BDF . Find the ratio ofthe area of circle P to the area of rectangle ABDE .
   √
   √
   π 3
   Answer: Let the side length of the hexagon be s . The length of BD is s 3, so the area
   √ √12
   of rectangle ABDE is s 3. Equilateral triangle BDF has side length s 3. The inradius of an 2
   √
   equilateral triangle is 3 / 6 times the length of its side, and so has length . Thus, the area of circles
   √2
   πs / 422
   πs π 3
   √
   P is , so the ratio is = .
   4 122
   s 3