Problem
1992 AHSME Problem 26
![[asy] fill((1,0)--arc((1,0),2,180,225)--cycle,grey); fill((-1,0)--arc((-1,0),2,315,360)--cycle,grey); fill((0,-1)--arc((0,-1),2-sqrt(2),225,315)--cycle,grey); fill((0,0)--arc((0,0),1,180,360)--cycle,white); draw((1,0)--arc((1,0),2,180,225)--(1,0),black+linewidth(1)); draw((-1,0)--arc((-1,0),2,315,360)--(-1,0),black+linewidth(1)); draw((0,0)--arc((0,0),1,180,360)--(0,0),black+linewidth(1)); draw(arc((0,-1),2-sqrt(2),225,315),black+linewidth(1)); draw((0,0)--(0,-1),black+linewidth(1)); MP("C",(0,0),N);MP("A",(-1,0),N);MP("B",(1,0),N); MP("D",(0,-.8),NW);MP("E",(1-sqrt(2),-sqrt(2)),SW);MP("F",(-1+sqrt(2),-sqrt(2)),SE); [/asy]](/static/aops/9a9d35d686ef3d3e0b131f2bf8575aa9b95ae828.png)
Semicircle \widehat{AB} has center C and radius 1. Point D is on \widehat{AB} and \overline{CD}\perp\overline{AB}. Extend \overline{BD} and \overline{AD} to E and F, respectively, so that circular arcs \widehat{AE} and \widehat{BF} have B and A as their respective centers. Circular arc \widehat{EF} has center D. The area of the shaded "smile" AEFBDA, is
\begin{array}{l} \text{(A) } \left(2-\sqrt{2}\right)\pi\quad \text{(B) } 2\pi-\pi \sqrt{2}-1\quad \text{(C) } \left(1-\frac{\sqrt{2}}{2}\right)\pi\quad\\ \text{(D) } \frac{5\pi}{2}-\pi\sqrt{2}-1\quad \text{(E) } \left(3-2\sqrt{2}\right)\pi \end{array}
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