PUMaC 2008 · 几何（B 组） · 第 3 题

PUMaC 2008 — Geometry (Division B) — Problem 3

(3 points) Consider a convex polygon P in space with perimeter 20 and area 30. What is the volume of the locus of points that are at most 1 unit away from some point in the interior of P ?

解析

Consider a convex polygon P in space with perimeter 20 and area 30. What is the volume of the locus of points that are at most 1 unit away from some point in the interior of P ? 34 ( ANS: 60 + π 3 The region is split into three sections: The volume directly above and below the convex polygon, π with volume 2 · 30 = 60, A half-cylinder of radius 1 at each edge, of total volume · 20, and a 2 4 sector of a sphere at each corner, of total volume π . So, the total volume is 3 4 34 60 + 10 π + π = 60 + π . CB: GL) 3 3