document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1103080>Question 1103080</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A circle is inscribed inside a regular hexagon. Find the ratio of the area of the circle to the area of the hexagon. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #717766 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=717766\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.1%2C1.1%2C-1.1%2C1.1%2C%0D%0Acircle%280%2C0%2C0.866%29%2Cgreen%28triangle%280%2C0%2C0%2C-0.866%2C0.5%2C-0.866%29%29%2C%0D%0Agreen%28rectangle%280%2C-0.866%2C0.1%2C-0.766%29%29%2Ctriangle%28-0.5%2C-0.866%2C0.5%2C-0.866%29%2C%0D%0Atriangle%28-0.5%2C0.866%2C0.5%2C0.866%29%2Ctriangle%28-1%2C0%2C-0.5%2C-0.866%2C0%2C0%29%2C%0D%0Atriangle%28-1%2C0%2C0%2C0%2C-0.5%2C0.866%29%2Ctriangle%281%2C0%2C0%2C0%2C0.5%2C0.866%29%2C%0D%0Alocate%280.02%2C-0.4%2CR%29%2Ctriangle%280%2C0%2C1%2C0%2C0.5%2C-0.866%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" >= radius of the circle = height of each of the 6 triangles forming the hexagon.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E2 ALIGN=MIDDLE  ALT=\"pi%2AR%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >= area of the circle<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%283%29R%5E2 ALIGN=MIDDLE  ALT=\"2sqrt%283%29R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >= area of the hexagon<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E2%2F%282sqrt%283%29R%5E2%29=highlight%28pi%2F%282sqrt%283%29%29=sqrt%283%29pi%2F6%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > is the ratio asked for.<BR>\n" );
document.write( " <BR>\n" );
document.write( "You may have a formula to calculate the area of a regular hexagon<BR>\n" );
document.write( "as a function of its apothem (which in this case is R),<BR>\n" );
document.write( "but I calculated it by figuring the sides and area of the triangles.<BR>\n" );
document.write( "The angles of those triangles measure <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60%5Eo ALIGN=MIDDLE  ALT=\"60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That makes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2860%5Eo%29=sqrt%283%29%2F2=R%2Fside ALIGN=MIDDLE  ALT=\"sin%2860%5Eo%29=sqrt%283%29%2F2=R%2Fside\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the triangles,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=side=2R%2Fsqrt%283%29=2sqrt%283%29R%2F3 ALIGN=MIDDLE  ALT=\"side=2R%2Fsqrt%283%29=2sqrt%283%29R%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Area for 1 triangle would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%282sqrt%283%29R%2F3%29R=sqrt%283%29R%5E2%2F3 ALIGN=MIDDLE  ALT=\"%281%2F2%29%282sqrt%283%29R%2F3%29R=sqrt%283%29R%5E2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the area of the hexagon is 6 times that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%28sqrt%283%29R%5E2%2F3%29=2sqrt%283%29R%5E2 ALIGN=MIDDLE  ALT=\"6%28sqrt%283%29R%5E2%2F3%29=2sqrt%283%29R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );