Math Prize 2012 Problem 2
Source:
September 24, 2012
geometry
Problem Statement
In the figure below, the centers of the six congruent circles form a regular hexagon with side length 2.
[asy]
import graph;
pair A = 2dir(0);
pair B = 2dir(60);
pair C = 2dir(120);
pair D = 2dir(180);
pair E = 2dir(240);
pair F = 2dir(300);
path hexagon = A -- B -- C -- D -- E -- F -- cycle;
fill(hexagon, gray);
filldraw(Circle(A, 1), white);
filldraw(Circle(B, 1), white);
filldraw(Circle(C, 1), white);
filldraw(Circle(D, 1), white);
filldraw(Circle(E, 1), white);
filldraw(Circle(F, 1), white);
draw(hexagon);
[/asy]
Adjacent circles are tangent to each other. What is the area of the shaded region?