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 #848582 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=848582\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Here is another way to solve this problem, this time getting an exact answer.<br><BR>\n" );
document.write( "Add 45-45-90 right triangles to alternating sides of the regular octagon to form a square.<br><BR>\n" );
document.write( "The hypotenuses of those triangles are edges of the regular octagon, so they each have edge length 1;  and so their legs all have length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The side length of the square is then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B2%28sqrt%282%29%2F2%29=1%2Bsqrt%282%29 ALIGN=MIDDLE  ALT=\"1%2B2%28sqrt%282%29%2F2%29=1%2Bsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Put the four added triangles together with their right angles at a common point to see that the combined area of the four triangles is the area of a square with side length 1.<br><BR>\n" );
document.write( "Then the area of the regular octagon is the area of a square with side length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Bsqrt%282%29 ALIGN=MIDDLE  ALT=\"1%2Bsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, minus the are of a square with side length 1:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bsqrt%282%29%29%5E2-1%5E2=%281%2B2sqrt%282%29%2B2%29-1=2%2B2sqrt%282%29 ALIGN=MIDDLE  ALT=\"%281%2Bsqrt%282%29%29%5E2-1%5E2=%281%2B2sqrt%282%29%2B2%29-1=2%2B2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2B2sqrt%282%29 ALIGN=MIDDLE  ALT=\"2%2B2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <<== typo corrected thanks to note from tutor @ikleyn<br><BR>\n" );
document.write( "(which to several decimal places is equal to the answer obtained by the other tutor, 4.828417...)<br><BR>\n" );
document.write( "NOTE: This problem shows a formula that is familiar to many geometry students who participate in math contests: the area of a regular octagon with side length s is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=s%5E2%282%2B2sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"A=s%5E2%282%2B2sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <<== typo corrected thanks to note from tutor @ikleyn<br><BR>\n" );
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