{"status": "success", "data": {"description_md": "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 regioninside the hexagon but outside all of the semicircles?<br><center><img class=\"problem-image\" alt='[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(\"$2$\",(3.5,3sqrt(3)/2),NE);  [/asy]' class=\"latexcenter\" height=\"205\" src=\"https://latex.artofproblemsolving.com/4/1/d/41d2ef082ac9be6c166ace09f122eaf7d1ccb5e2.png\" width=\"235\"/></center>\n\n$\\textbf {(A) } 6\\sqrt{3}-3\\pi \\qquad \\textbf {(B) } \\frac{9\\sqrt{3}}{2} - 2\\pi\\ \\qquad \\textbf {(C) } \\frac{3\\sqrt{3}}{2} - \\frac{\\pi}{3} \\qquad \\textbf {(D) } 3\\sqrt{3} - \\pi \\qquad \\textbf {(E) } \\frac{9\\sqrt{3}}{2} - \\pi$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length  <span class=\"katex--inline\">2</span>  so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region&#8212;inside the hexagon but outside all of the semicircles?<br/><center><img class=\"latexcenter\" alt=\"[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(&#34;$2$&#34;,(3.5,3sqrt(3)/2),NE);  [/asy]\" height=\"205\" src=\"https://latex.artofproblemsolving.com/4/1/d/41d2ef082ac9be6c166ace09f122eaf7d1ccb5e2.png\" width=\"235\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf {(A) } 6\\sqrt{3}-3\\pi \\qquad \\textbf {(B) } \\frac{9\\sqrt{3}}{2} - 2\\pi\\ \\qquad \\textbf {(C) } \\frac{3\\sqrt{3}}{2} - \\frac{\\pi}{3} \\qquad \\textbf {(D) } 3\\sqrt{3} - \\pi \\qquad \\textbf {(E) } \\frac{9\\sqrt{3}}{2} - \\pi</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2020 AMC 12B Problem 11", "can_next": true, "can_prev": true, "nxt": "/problem/20_amc12B_p12", "prev": "/problem/20_amc12B_p10"}}