MathDB

A circle of radius 2 is centered at

O

. Square

OABC

has side length 1. Sides

\overline{AB}

and

\overline{CB}

are extended past

b

to meet the circle at

D

and

E

, respectively. What is the area of the shaded region in the figure, which is bounded by

\overline{BD}

\overline{BE}

, and the minor arc connecting

D

and

E

?
[asy] defaultpen(linewidth(0.8)); pair O=origin, A=(1,0), C=(0,1), B=(1,1), D=(1, sqrt(3)), E=(sqrt(3), 1), point=B; fill(Arc(O, 2, 0, 90)--O--cycle, mediumgray); clip(B--Arc(O, 2, 30, 60)--cycle); draw(Circle(origin, 2)); draw((-2,0)--(2,0)^^(0,-2)--(0,2)); draw(A--D^^C--E); label("

A

", A, dir(point--A)); label("

C

", C, dir(point--C)); label("

O

", O, dir(point--O)); label("

D

", D, dir(point--D)); label("

E

", E, dir(point--E)); label("

B

", B, SW);[/asy]
(A) \frac {\pi}3 \plus{} 1 \minus{} \sqrt {3} \qquad (B) \frac {\pi}2\left( 2 \minus{} \sqrt {3}\right) \qquad (C) \pi\left(2 \minus{} \sqrt {3}\right) \qquad (D) \frac {\pi}{6} \plus{} \frac {\sqrt {3} \minus{} 1}{2} \\ \qquad \indent (E) \frac {\pi}{3} \minus{} 1 \plus{} \sqrt {3}

Problem Statement