document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=193354>Question 193354</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the area of a regular hexagon with a perimeter of 90 centimeters </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #145152 by </B><A HREF=/tutors/aboutme.mpl?userid=RAY100><B>RAY100(1637)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=RAY100><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=145152\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> very  complicated,,,please  start  with  a  sketch,  or  refer  to  geometry book  under  area  polygons\r<BR>\n" );
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document.write( "regular  hexagon  is   six  sided,  with  all  sides  equal.<BR>\n" );
document.write( "if  we  make  a  rough  sketch,  and  then  find  center  roughly.<BR>\n" );
document.write( "from  center  draw  a  line  to  each  of  vertex,,,,should  be  six  triangles.<BR>\n" );
document.write( "the  central  angle  of  each  triangle  should  be  120  deg,,,,360/6<BR>\n" );
document.write( "now  draw  an  altitude  on  just  one  of  the  triangles,,,,it  should  go  from  center  to  opposite  base.<BR>\n" );
document.write( "It  bisects  the  base,  and  is  perpendicular.<BR>\n" );
document.write( "That  smaller  triangle  should  have  angles  of  90(at  base),60(at  center,  half  of  120),  and  30(180-60-90)\r<BR>\n" );
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document.write( "Remember  that  a  30  -60-90  triangle  have  sides  proportional  to, 1-2-sqrt3<BR>\n" );
document.write( "the  base  side  is  90/6=15 long,<BR>\n" );
document.write( "the  base  of  the  small  triangle  is  half of  15 or 7.5  degrees<BR>\n" );
document.write( "this  7.5  side  is  proportional  to  the  sqrt3  typical  side<BR>\n" );
document.write( "the  height  is  proportional  to  1,  on  the  typical  triangle.\r<BR>\n" );
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document.write( "(sqrt3/1)=(7.5/h)<BR>\n" );
document.write( "cross  mult<BR>\n" );
document.write( "sqrt3 *h=7.5<BR>\n" );
document.write( "h=7.5/sqrt3=4.333\r<BR>\n" );
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document.write( "now  the  area  of  any  triangle  is  (1/2)base *height<BR>\n" );
document.write( "the  larger  triangle  has  an  area  of  (1/2) 15*4.33 =  32.475<BR>\n" );
document.write( "but  a  hexagon  is  six  large  triangles,  6*32.475=194.85\r<BR>\n" );
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document.write( "in  general  the  area  of  any  polygon  is  1/2 * perimeter*apothem<BR>\n" );
document.write( "apothem (a)  is  fancy  word  for  the  height  above,  perp  distance  from  base  to  center<BR>\n" );
document.write( "in  our  case,  A  =1/2*90*4.33=194.85\r<BR>\n" );
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document.write( "normally,  in  area  of  polygon  problems,  the  apothem  is  given </SPAN>\n" );
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