document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=568980>Question 568980</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the perimeter of a regular dodecagon (a polygon which has 12 sides)whose area is 24 + 12 sqrt(3). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #367338 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=367338\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To calculate the area of regular polygon with N sides, the polygon can be sliced pizza-style into N isosceles triangles with one side as the base, and the area of those triangles can be calculated.<BR>\n" );
document.write( "The height of those triangles is called the apothem, and can be calculated based on the length of the side, s, and half of the vertex angle, which depends on the number of sides, N.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C160%2C-2%2C28%2C-8%2C8%2C%0D%0Atriangle%280%2C7%2C0%2C-7%2C26.124%2C0%29%2C%0D%0Aline%280%2C0%2C26.124%2C0%29%2C%0D%0Alocate%288%2C0%2Capothem%29%2C+locate%28-1.2%2C1.1%2Cs%29%2Clocate%2820%2C1.5%2CB%29%2C%0D%0Aarrow%28-1%2C0.8%2C-1%2C7%29%2Carrow%28-1%2C-0.8%2C-1%2C-7%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > For a dodecagon, the half of the vertex angle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=360%5Eo%2F24 ALIGN=MIDDLE  ALT=\"B=360%5Eo%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Putting it all together, the area, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >, of a regular polygon with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N ALIGN=MIDDLE  ALT=\"N\"   DISABLED_style= \"max-width:90%; height:auto;\" > sides of length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s ALIGN=MIDDLE  ALT=\"s\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=s%5E2N%2F%284tan%28pi%2Fn%29%29 ALIGN=MIDDLE  ALT=\"A=s%5E2N%2F%284tan%28pi%2Fn%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=s%5E2N%2F%284tan%28180%5Eo%2Fn%29%29 ALIGN=MIDDLE  ALT=\"A=s%5E2N%2F%284tan%28180%5Eo%2Fn%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > if you prefer your angles in degrees.<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N=12 ALIGN=MIDDLE  ALT=\"N=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=s%5E2%2A12%2F%284tan%28180%5Eo%2F12%29%29=3s%5E2%2Ftan%2815%5Eo%29 ALIGN=MIDDLE  ALT=\"A=s%5E2%2A12%2F%284tan%28180%5Eo%2F12%29%29=3s%5E2%2Ftan%2815%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Atan%2815%5Eo%29=3s%5E2 ALIGN=MIDDLE  ALT=\"Atan%2815%5Eo%29=3s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2=Atan%2815%5Eo%29%2F3 ALIGN=MIDDLE  ALT=\"s%5E2=Atan%2815%5Eo%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2815%5Eo%29=2-sqrt%283%29 ALIGN=MIDDLE  ALT=\"tan%2815%5Eo%29=2-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as can be calculated from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2845%5Eo%29 ALIGN=MIDDLE  ALT=\"tan%2845%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2830%5Eo%29 ALIGN=MIDDLE  ALT=\"tan%2830%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the formula (trigonometric identity) for tangent of a difference.<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2=Atan%2815%5Eo%29%2F3 ALIGN=MIDDLE  ALT=\"s%5E2=Atan%2815%5Eo%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=24%2B12sqrt%283%29=12%282%2Bsqrt%283%29%29 ALIGN=MIDDLE  ALT=\"A=24%2B12sqrt%283%29=12%282%2Bsqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2815%5Eo%29=2-sqrt%283%29 ALIGN=MIDDLE  ALT=\"tan%2815%5Eo%29=2-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives us<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2=12%282%2Bsqrt%283%29%29%282-sqrt%283%29%29%2F3=12%282%5E2-%28sqrt%283%29%29%5E2%29%2F3=4%284-3%29=4%2A1=4 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The length of a side of the polygon is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=2 ALIGN=MIDDLE  ALT=\"s=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and the perimeter is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A2=highlight%2824%29 ALIGN=MIDDLE  ALT=\"12%2A2=highlight%2824%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );