PUMaC 2013 · 几何（A 组） · 第 3 题

PUMaC 2013 — Geometry (Division A) — Problem 3

[ 4 ] Consider the shape formed from taking equilateral triangle ABC with side length 6 and tracing out the arc BC with center A . Set the shape down on line l so that segment AB is perpendicular to l , and B touches l . Beginning from arc BC touching l , We roll ABC along l until both points A and C are on the line. The area traced out by the roll can be written in the form nπ , where n is an integer. Find n . ◦ ◦ ◦

解析

[ 4 ] Consider the shape formed from taking equilateral triangle ABC with side length 6 and tracing out the arc BC with center A . Set the shape down on line l so that segment AB is perpendicular to l , and B touches l . Beginning from arc BC touching l , We roll ABC along l until both points A and C are on the line. The area traced out by the roll can be written in the form n ⇡ , where n is an integer. Find n . Solution The roll can be broken down into two parts. The first part, when the shape rolls along arc BC , traces a rectangle of area 12 ⇡ . The second part, when the shape rolls about the point C , traces a quarter circle of area 9 ⇡ . The total area is 21 ⇡ .