document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1113612>Question 1113612</A>: <I><SPAN STYLE=\"word-break: break-all;\"> So I have this dodecagon with the side length of 2, it is inscribed in a square. How do i find the area of the square with only that information? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #728680 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=728680\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Obviously you mean a regular dodecagon; otherwise the problem can't be solved....<br><BR>\n" );
document.write( "Let A, B, C, D, and E be consecutive vertices of the dodecagon; with sides AB and DE on adjacent sides of the square.<br><BR>\n" );
document.write( "Let F be the corner of the square determined by the extension of sides AB and DE of the dodecagon; that is, AF and FE are portions of adjacent sides of the square.<br><BR>\n" );
document.write( "Draw segments from vertex C of the dodecagon parallel to AF and FE, forming a small square in the corner of the big square.  The area outside the dodecagon and inside the square now consists of that small square and two small right triangles.<br><BR>\n" );
document.write( "The exterior angle of a regular dodecagon is 30 degrees; that means the small right triangles are 30-60-90 right triangles.  The hypotenuse of each of those triangles is a side of the dodecagon, with length 2.  Then the lengths of the legs of the triangles are 1 and sqrt(3).<br><BR>\n" );
document.write( "Then by symmetry the full side length of the large square is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Bsqrt%283%29%2B2%2Bsqrt%283%29%2B1+=+4%2B2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"1%2Bsqrt%283%29%2B2%2Bsqrt%283%29%2B1+=+4%2B2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then the area of the square is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2B2%2Asqrt%283%29%29%5E2+=+28%2B16%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"%284%2B2%2Asqrt%283%29%29%5E2+=+28%2B16%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "----------------------------------------------------------<BR>\n" );
document.write( "<br><BR>\n" );
document.write( "Alternate solution method....<br><BR>\n" );
document.write( "Trigonometry makes this easy.<br><BR>\n" );
document.write( "Let AB be a side of the dodecagon which lies on a side of the square, and consider the triangle OAB where O is the center of the figure (center of both the square and the dodecagon).<br><BR>\n" );
document.write( "If M is the midpoint of AB, then OM is an apothem of the dodecagon; and the length of the apothem is half the side of the square.<br><BR>\n" );
document.write( "Angle AOB is 30 degrees, so angle AOM is 15 degrees.<br><BR>\n" );
document.write( "With AB being 2, AM is 1, so the length of the apothem is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%2815%29 ALIGN=MIDDLE  ALT=\"cot%2815%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Then the side length of the square is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Acot%2815%29+=+2%282%2Bsqrt%283%29%29+=+4%2B2%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"2%2Acot%2815%29+=+2%282%2Bsqrt%283%29%29+=+4%2B2%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, as before. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );