Four Semicircles
Source:
January 5, 2009
geometrytrigonometry
Problem Statement
Three semicircles of radius are constructed on diameter of a semicircle of radius . The centers of the small semicircles divide into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?
[asy]import graph;
unitsize(14mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dashed=linetype("4 4");
dotfactor=3;pair A=(-2,0), B=(2,0);fill(Arc((0,0),2,0,180)--cycle,mediumgray);
fill(Arc((-1,0),1,0,180)--cycle,white);
fill(Arc((0,0),1,0,180)--cycle,white);
fill(Arc((1,0),1,0,180)--cycle,white);
draw(Arc((-1,0),1,60,180));
draw(Arc((0,0),1,0,60),dashed);
draw(Arc((0,0),1,60,120));
draw(Arc((0,0),1,120,180),dashed);
draw(Arc((1,0),1,0,120));
draw(Arc((0,0),2,0,180)--cycle);dot((0,0));
dot((-1,0));
dot((1,0));draw((-2,-0.1)--(-2,-0.3),gray);
draw((-1,-0.1)--(-1,-0.3),gray);
draw((1,-0.1)--(1,-0.3),gray);
draw((2,-0.1)--(2,-0.3),gray);label("",A,W);
label("",B,E);
label("1",(-1.5,-0.1),S);
label("2",(0,-0.1),S);
label("1",(1.5,-0.1),S);[/asy] (A)\ \pi\minus{}\sqrt3 \qquad
(B)\ \pi\minus{}\sqrt2 \qquad
(C)\ \frac{\pi\plus{}\sqrt2}{2} \qquad
(D)\ \frac{\pi\plus{}\sqrt3}{2}
(E)\ \frac{7}{6}\pi\minus{}\frac{\sqrt3}{2}