|
(Tag: Redirect target changed) |
| (13 intermediate revisions by 4 users not shown) |
| Line 1: |
Line 1: |
| − | ==Problem==
| + | #REDIRECT [[2020 AMC 10B Problems/Problem 14]] |
| − | As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region—inside the hexagon but outside all of the semicircles?
| |
| − | | |
| − | [asy] | |
| − | size(140);
| |
| − | fill((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--cycle,gray(0.4));
| |
| − | fill(arc((2,0),1,180,0)--(2,0)--cycle,white);
| |
| − | fill(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle,white);
| |
| − | fill(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle,white);
| |
| − | fill(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle,white);
| |
| − | fill(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle,white);
| |
| − | fill(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle,white);
| |
| − | draw((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--(1,0));
| |
| − | draw(arc((2,0),1,180,0)--(2,0)--cycle);
| |
| − | draw(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle);
| |
| − | draw(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle);
| |
| − | draw(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle);
| |
| − | draw(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle);
| |
| − | draw(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle);
| |
| − | label("<math>2</math>",(3.5,3sqrt(3)/2),NE);
| |
| − | [/asy] | |
| − | | |
| − | <math>\textbf{(A)}\ 6\sqrt3-3\pi \qquad\textbf{(B)}\ \frac{9\sqrt3}{2}-2\pi \qquad\textbf{(C)}\ \frac{3\sqrt3}{2}-\frac{\pi}{3} \qquad\textbf{(D)}\ 3\sqrt3-\pi \\ \qquad\textbf{(E)}\ \frac{9\sqrt3}{2}-\pi</math>
| |
| − | | |
| − | ==Solution==
| |
| − | | |
| − | ==See Also==
| |
| − | | |
| − | {{AMC12 box|year=2020|ab=B|num-b=10|num-a=12}}
| |
| − | {{MAA Notice}}
| |
