HMMT 二月 2014 · 冲刺赛 · 第 3 题

HMMT February 2014 — Guts Round — Problem 3

[ 4 ] Let ABCDEF be a regular hexagon. Let P be the circle inscribed in 4 BDF . Find the ratio of the area of circle P to the area of rectangle ABDE .

解析

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