{"status": "success", "data": {"description_md": "The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\\frac{2\\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve? <br><center><img class=\"problem-image\" alt=\"[asy] size(6cm); defaultpen(fontsize(6pt)); dotfactor=4; label(&quot;$\\circ$&quot;,(0,1)); label(&quot;$\\circ$&quot;,(0.865,0.5)); label(&quot;$\\circ$&quot;,(-0.865,0.5)); label(&quot;$\\circ$&quot;,(0.865,-0.5)); label(&quot;$\\circ$&quot;,(-0.865,-0.5)); label(&quot;$\\circ$&quot;,(0,-1)); dot((0,1.5)); dot((-0.4325,0.75)); dot((0.4325,0.75)); dot((-0.4325,-0.75)); dot((0.4325,-0.75)); dot((-0.865,0)); dot((0.865,0)); dot((-1.2975,-0.75)); dot((1.2975,-0.75)); draw(Arc((0,1),0.5,210,-30)); draw(Arc((0.865,0.5),0.5,150,270)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0,-1),0.5,30,150)); draw(Arc((-0.865,-0.5),0.5,330,90)); draw(Arc((-0.865,0.5),0.5,-90,30)); [/asy]\" class=\"latexcenter\" height=\"265\" src=\"https://latex.artofproblemsolving.com/b/8/8/b887867827b6b53efb70abc6807aa51c7a2287fe.png\" width=\"285\"/></center>\n\n$\\textbf{(A)}\\ 2\\pi+6\\qquad\\textbf{(B)}\\ 2\\pi+4\\sqrt3 \\qquad\\textbf{(C)}\\ 3\\pi+4 \\qquad\\textbf{(D)}\\ 2\\pi+3\\sqrt3+2 \\qquad\\textbf{(E)}\\ \\pi+6\\sqrt3$\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>The closed curve in the figure is made up of  <span class=\"katex--inline\">9</span>  congruent circular arcs each of length  <span class=\"katex--inline\">\\frac{2\\pi}{3}</span> , where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side  <span class=\"katex--inline\">2</span> . What is the area enclosed by the curve? <br/><center><img class=\"latexcenter\" alt=\"[asy] size(6cm); defaultpen(fontsize(6pt)); dotfactor=4; label(&#34;$\\circ$&#34;,(0,1)); label(&#34;$\\circ$&#34;,(0.865,0.5)); label(&#34;$\\circ$&#34;,(-0.865,0.5)); label(&#34;$\\circ$&#34;,(0.865,-0.5)); label(&#34;$\\circ$&#34;,(-0.865,-0.5)); label(&#34;$\\circ$&#34;,(0,-1)); dot((0,1.5)); dot((-0.4325,0.75)); dot((0.4325,0.75)); dot((-0.4325,-0.75)); dot((0.4325,-0.75)); dot((-0.865,0)); dot((0.865,0)); dot((-1.2975,-0.75)); dot((1.2975,-0.75)); draw(Arc((0,1),0.5,210,-30)); draw(Arc((0.865,0.5),0.5,150,270)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0.865,-0.5),0.5,90,-150)); draw(Arc((0,-1),0.5,30,150)); draw(Arc((-0.865,-0.5),0.5,330,90)); draw(Arc((-0.865,0.5),0.5,-90,30)); [/asy]\" height=\"265\" src=\"https://latex.artofproblemsolving.com/b/8/8/b887867827b6b53efb70abc6807aa51c7a2287fe.png\" width=\"285\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 2\\pi+6\\qquad\\textbf{(B)}\\ 2\\pi+4\\sqrt3 \\qquad\\textbf{(C)}\\ 3\\pi+4 \\qquad\\textbf{(D)}\\ 2\\pi+3\\sqrt3+2 \\qquad\\textbf{(E)}\\ \\pi+6\\sqrt3</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": "2012 AMC 12A Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12A_p15", "prev": "/problem/12_amc12A_p13"}}