Area of the Circle in the Intersection of Two Circles
Source:
January 15, 2009
geometrytrigonometry
Problem Statement
A circle of radius is internally tangent to two circles of radius at points and , where is a diameter of the smaller circle. What is the area of the region, shaded in the gure, that is outside the smaller circle and inside each of the two larger circles?
[asy]size(200);defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;pair B = (0,1);
pair A = (0,-1);label("",B,NW);label("",A,2S);draw(Circle(A,2));draw(Circle(B,2));fill((-sqrt(3),0)..B..(sqrt(3),0)--cycle,gray);
fill((-sqrt(3),0)..A..(sqrt(3),0)--cycle,gray);
draw((-sqrt(3),0)..B..(sqrt(3),0));
draw((-sqrt(3),0)..A..(sqrt(3),0));path circ = Circle(origin,1);fill(circ,white);
draw(circ);
dot(A);dot(B);pair A1 = B + dir(45)*2;
pair A2 = dir(45);
pair A3 = dir(-135)*2 + A;draw(B--A1,EndArrow(HookHead,2));
draw(origin--A2,EndArrow(HookHead,2));
draw(A--A3,EndArrow(HookHead,2)); label("",midpoint(B--A1),NW);
label("",midpoint(origin--A2),NW);
label("",midpoint(A--A3),NW);[/asy] (A)\ \frac {5}{3}\pi \minus{} 3\sqrt {2}\qquad (B)\ \frac {5}{3}\pi \minus{} 2\sqrt {3}\qquad (C)\ \frac {8}{3}\pi \minus{} 3\sqrt {3}\qquad(D)\ \frac {8}{3}\pi \minus{} 3\sqrt {2}
(E)\ \frac {8}{3}\pi \minus{} 2\sqrt {3}