Lune
Source:
February 19, 2008
geometrycalculusanalytic geometryintegrationLaTeX
Problem Statement
A semicircle of diameter sits at the top of a semicircle of diameter , as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.
[asy]unitsize(2.5cm);
defaultpen(fontsize(10pt)+linewidth(.8pt));filldraw(Circle((0,.866),.5),grey,black);
label("1",(0,.866),S);
filldraw(Circle((0,0),1),white,black);
draw((-.5,.866)--(.5,.866),linetype("4 4"));
clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle);
draw((-1,0)--(1,0));
label("2",(0,0),S);[/asy] (A)\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad (B)\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad (C)\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad(D)\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi
(E)\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi