document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209347>Question 1209347</A>: <I><SPAN STYLE=\"word-break: break-all;\"> We cut a regular octagon ABCDEFGH out of a piece of cardboard.  If $AB = 1$, then what is the area of the octagon?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848577 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=848577\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Draw a segment from the center to each vertex of the octagon. <BR>\n" );
document.write( "It will produce 8 pizza slices so to speak.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C0%2C3%2C0.75%2C3.75%2C%0D%0A%0D%0Aline%281%2C1%2C2%2C1%29%2Cline%282%2C1%2C2.70711%2C1.70711%29%2Cline%282.70711%2C1.70711%2C2.70711%2C2.70711%29%2Cline%282.70711%2C2.70711%2C2%2C3.41421%29%2Cline%282%2C3.41421%2C1%2C3.41421%29%2Cline%281%2C3.41421%2C0.29289%2C2.70711%29%2Cline%280.29289%2C2.70711%2C0.29289%2C1.70711%29%2Cline%280.29289%2C1.70711%2C1%2C1%29%2C%0D%0A%0D%0Aline%281.5%2C2.20711%2C1%2C1%29%2Cline%281.5%2C2.20711%2C2%2C1%29%2Cline%281.5%2C2.20711%2C2.70711%2C1.70711%29%2Cline%281.5%2C2.20711%2C2.70711%2C2.70711%29%2Cline%281.5%2C2.20711%2C2%2C3.41421%29%2Cline%281.5%2C2.20711%2C1%2C3.41421%29%2Cline%281.5%2C2.20711%2C0.29289%2C2.70711%29%2Cline%281.5%2C2.20711%2C0.29289%2C1.70711%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Each pizza slice is an isosceles triangle with an apex angle of 360/8 = 45 degrees\r<BR>\n" );
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document.write( "Let's zoom in on one of the pizza slices. <BR>\n" );
document.write( "I'll draw a line to split this slice into two mirror halves.<BR>\n" );
document.write( "This splits the 45 degree angle into 45/2 = 22.5 degrees.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C0%2C3%2C0.75%2C3.75%2C%0D%0Aline%280.3%2C1%2C2.7%2C1%29%2Cline%282.7%2C1%2C1.5%2C3.4%29%2Cline%281.5%2C3.4%2C0.3%2C1%29%2Cline%281.5%2C3.4%2C1.5%2C1%29%2C%0D%0Alocate%281.4%2C2%2C%22h%22%29%2Clocate%281.9%2C0.95%2C%220.5%22%29%2Clocate%280.9%2C0.95%2C%220.5%22%29%2Clocate%281.55%2C2.6%2C22.5%5Eo%29%2Clocate%281.15%2C2.6%2C22.5%5Eo%29%2C%0D%0Aline%281.5%2C1.2%2C1.7%2C1.2%29%2Cline%281.7%2C1.2%2C1.7%2C1%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "h = unknown height of this pizza slice<BR>\n" );
document.write( "h = apothem of the octagon<BR>\n" );
document.write( "The side length of the octagon has been split in half to 1/2 = 0.5\r<BR>\n" );
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document.write( "tan(angle) = opposite/adjacent<BR>\n" );
document.write( "tan(22.5) = 0.5/h<BR>\n" );
document.write( "h*tan(22.5) = 0.5<BR>\n" );
document.write( "h = 0.5/tan(22.5)<BR>\n" );
document.write( "h = 1.207106781187 approximately<BR>\n" );
document.write( "Please make sure that your calculator is set to degrees mode.\r<BR>\n" );
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document.write( "The area of each of the original slices would be<BR>\n" );
document.write( "area = 0.5*base*height<BR>\n" );
document.write( "area = 0.5*1*1.207106781187<BR>\n" );
document.write( "area = 0.603553390593<BR>\n" );
document.write( "8 times of which leads to a total area of 8*0.603553390593 = <font color=red>4.828427125</font> which is approximate.\r<BR>\n" );
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document.write( "Another approach would be to use this formula<BR>\n" );
document.write( "Area = 0.25*s^2*n*cot(180/n)<BR>\n" );
document.write( "where \"cot\" represents \"cotangent\". Recall that cot = 1/tan<BR>\n" );
document.write( "s = side length<BR>\n" );
document.write( "n = number of sides\r<BR>\n" );
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document.write( "If we use this formula then we'll have s = 1 and n = 8<BR>\n" );
document.write( "Area = 0.25*s^2*n*cot(180/n)<BR>\n" );
document.write( "Area = 0.25*1^2*8*cot(180/8)<BR>\n" );
document.write( "Area = 2*cot(22.5)<BR>\n" );
document.write( "Area = 2*1/tan(22.5)<BR>\n" );
document.write( "Area = <font color=red>4.828427125</font>\r<BR>\n" );
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document.write( "Edit: after greenestamps made his post I realize that tan(22.5) can be determined exactly in terms of radicals.\r<BR>\n" );
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document.write( "Use the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%2F2%29+=+%22%22%2B-sqrt%28+%281-cos%28theta%29%29%2F%281%2Bcos%28theta%29%29%29 ALIGN=MIDDLE  ALT=\"tan%28theta%2F2%29+=+%22%22%2B-sqrt%28+%281-cos%28theta%29%29%2F%281%2Bcos%28theta%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to determine that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2845%2F2%29+=+sqrt%28+%281-cos%2845%29%29%2F%281%2Bcos%2845%29%29%29 ALIGN=MIDDLE  ALT=\"tan%2845%2F2%29+=+sqrt%28+%281-cos%2845%29%29%2F%281%2Bcos%2845%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Note that tan(22.5) is positive so we ignore the minus outside the root.<BR>\n" );
document.write( "After simplification you should get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%2822.5%29+=+-1%2Bsqrt%282%29 ALIGN=MIDDLE  ALT=\"tan%2822.5%29+=+-1%2Bsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cot%2822.5%29+=+1%2Bsqrt%282%29 ALIGN=MIDDLE  ALT=\"cot%2822.5%29+=+1%2Bsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So, area = 2*cot(22.5) = 2*(1+sqrt(2)) = <font color=red>1+2*sqrt(2)</font>\r<BR>\n" );
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document.write( "Exact Answer: <font color=red>2+2*sqrt(2) square units</font> <BR>\n" );
document.write( "Approximate Answer: <font color=red>4.828427125 square units</font><BR>\n" );
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