document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=78947>Question 78947</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please show me how to find the area of a regular hexagon with a perimeter of 100cm. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #56670 by </B><A HREF=/tutors/aboutme.mpl?userid=mathdoc314><B>mathdoc314(58)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathdoc314><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=56670\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It would be easier to draw this, but here goes - I will type it\r<BR>\n" );
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document.write( " P ___ Q\r\n" );
document.write( "U /___\ R\r\n" );
document.write( "  \___/\r\n" );
document.write( "T      S\r\n" );
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document.write( "Draw diagonals QT and PS too.<BR>\n" );
document.write( "Label the center O.<BR>\n" );
document.write( "This divides your regular hexagon into six equilateral triangles.<BR>\n" );
document.write( "These lines all have equal length:<BR>\n" );
document.write( " PQ, QR, RS, ST, TU,UP,PO,QO,RO,SO,TO,UO<BR>\n" );
document.write( "The perimiter is 100 (all lengths in cm here) so if s is the length of those lines then 6 s = 100, or s = 100/6 (cm)\r<BR>\n" );
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document.write( "There is a point halfway between S and T on line ST (the midpoint).<BR>\n" );
document.write( "Label this as M.  Consider the triangle OSM:\r<BR>\n" );
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document.write( "   O\r\n" );
document.write( "   |\\r\n" );
document.write( " h | \s\r\n" );
document.write( "   |  \\r\n" );
document.write( " M ---- S\r\n" );
document.write( "   s/2\r\n" );
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document.write( "Use the pythagorean theorem (or trig) to find out the missing height h:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+h%5E2+=+s%5E2+-+%28s%2F2%29%5E2+ ALIGN=MIDDLE  ALT=\"+h%5E2+=+s%5E2+-+%28s%2F2%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+h%5E2+=+3+s%5E2+%2F+4++ ALIGN=MIDDLE  ALT=\"+h%5E2+=+3+s%5E2+%2F+4++\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+h+=+%28sqrt%283%29%2F2%29+s+ ALIGN=MIDDLE  ALT=\"+h+=+%28sqrt%283%29%2F2%29+s+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now you know the area of triangle OMS is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1%2F2+%2A+%28sqrt%283%29%2F2%29+s+%2A+%28s%2F2%29+ ALIGN=MIDDLE  ALT=\"+1%2F2+%2A+%28sqrt%283%29%2F2%29+s+%2A+%28s%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (one half times the base times the height)\r<BR>\n" );
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document.write( "simplified, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28sqrt%283%29+%2A+s%5E2%29+%2F+8+ ALIGN=MIDDLE  ALT=\"+%28sqrt%283%29+%2A+s%5E2%29+%2F+8+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The area of the hexagon is 12 times this because the hexagon can be decomposed exactly into 12 triangles each congruent to OSM.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+12+%2A+%28%28sqrt%283%29+%2A+s%5E2%29+%2F+8%29+=+%28%283+%2A+sqrt%283%29%29+%2F+2%29+s%5E2+ 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=+A+=+%28%283+%2A+sqrt%283%29%29+%2F+2%29+%2A+%28100%2F6%29%5E2+=+10000+%2A+sqrt%283%29+%2F+24+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is approximately 721.688\r<BR>\n" );
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document.write( "As a check, compare to the area of the circumscribed circle of the hexagon<BR>\n" );
document.write( "pi * (100/6)^2 = 872.665\r<BR>\n" );
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document.write( "I would expect these two areas to be a little closer together.  You may want to check my work above.  But I think it is pretty much right.\r<BR>\n" );
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