document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1112228>Question 1112228</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two regular quadrilateral vinyl tiles each of 1 ft sides overlap each other such that the overlapping region is a regular octagon. What is the area of the overlapping region? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #727342 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=727342\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> FIRST A PICTURE (OR 2), TO VISUALIZE THE PROBLEM:<BR>\n" );
document.write( "A regular quadrilateral is called a square.<BR>\n" );
document.write( "The sketch below show red square tile<BR>\n" );
document.write( "overlapping a green square tile<BR>\n" );
document.write( "in the manner described in the problem.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.5%2C1.5%2C-1.5%2C1.5%2C%0D%0Agreen%28triangle%28-1.414%2C0%2C-1%2C0.414%2C-1%2C-0.414%29%29%2C%0D%0Agreen%28triangle%281.414%2C0%2C1%2C0.414%2C1%2C-0.414%29%29%2C%0D%0Agreen%28triangle%280%2C-1.414%2C0.414%2C-1%2C-0.414%2C-1%29%29%2C%0D%0Agreen%28triangle%280%2C1.414%2C0.414%2C1%2C-0.414%2C1%29%29%2C%0D%0Ared%28rectangle%28-1%2C-1%2C1%2C1%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The corners of the red tile are unsupported (not resting on the other tile).<BR>\n" );
document.write( "Let me cut them out, so you can see the regular octagon.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.5%2C1.5%2C-1.5%2C1.5%2C%0D%0Agreen%28triangle%28-1.414%2C0%2C-1%2C0.414%2C-1%2C-0.414%29%29%2C%0D%0Agreen%28triangle%281.414%2C0%2C1%2C0.414%2C1%2C-0.414%29%29%2C%0D%0Agreen%28triangle%280%2C-1.414%2C0.414%2C-1%2C-0.414%2C-1%29%29%2C%0D%0Agreen%28triangle%280%2C1.414%2C0.414%2C1%2C-0.414%2C1%29%29%2C%0D%0Ared%28rectangle%28-1%2C-1%2C1%2C1%29%29%2C%0D%0Aline%28-1%2C-0.414%2C-0.414%2C-1%29%2Cline%281%2C0.414%2C0.414%2C1%29%2C%0D%0Aline%28-1%2C0.414%2C-0.414%2C1%29%2Cline%281%2C-0.414%2C0.414%2C-1%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "CALCULATING THE AREA:<BR>\n" );
document.write( "You could calculate the area of one square, and subtract<BR>\n" );
document.write( "the area of the four corner triangles that are not overlapping the other square.<BR>\n" );
document.write( "You could use the formula for the area of a regular polygon,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=perimeter%2Aapothem%2F2 ALIGN=MIDDLE  ALT=\"area=perimeter%2Aapothem%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=%281%2F2%29%2Aperimeter%2Aapothem ALIGN=MIDDLE  ALT=\"area=%281%2F2%29%2Aperimeter%2Aapothem\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Either way, you need to know the length of a side of the octagon,<BR>\n" );
document.write( "or the sides of those corner triangles.<BR>\n" );
document.write( "Those corner triangles are isosceles right triangles,<BR>\n" );
document.write( "meaning the two short sides (legs) have the same lengths.<BR>\n" );
document.write( "Applying the Pythagorean theorem,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=leg%5E2%2Bleg%5E2=hypotenuse%5E2 ALIGN=MIDDLE  ALT=\"leg%5E2%2Bleg%5E2=hypotenuse%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2leg%5E2=hypotenuse%5E2 ALIGN=MIDDLE  ALT=\"2leg%5E2=hypotenuse%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%0D%0A%2Aleg=hypotenuse ALIGN=MIDDLE  ALT=\"sqrt%282%29%0D%0A%2Aleg=hypotenuse\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=leg=hypotenuse%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"leg=hypotenuse%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The side of the red square has a length of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > foot,<BR>\n" );
document.write( "so looking at one side of the red square and the attached triangles,<BR>\n" );
document.write( "we see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=leg%2Bhypotenuse%2Bleg=1foot ALIGN=MIDDLE  ALT=\"leg%2Bhypotenuse%2Bleg=1foot\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the length in ft of the side of the octagon (the hypotenuse of the triangles),<BR>\n" );
document.write( "we can re-write the equations above into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fsqrt%282%29%2Bx%2Bx%2Fsqrt%282%29=1 ALIGN=MIDDLE  ALT=\"x%2Fsqrt%282%29%2Bx%2Bx%2Fsqrt%282%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2x%2Fsqrt%282%29=1 ALIGN=MIDDLE  ALT=\"x%2B2x%2Fsqrt%282%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Bx%2Asqrt%282%29=1 ALIGN=MIDDLE  ALT=\"x%2Bx%2Asqrt%282%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%281%2Bsqrt%282%29%29=1 ALIGN=MIDDLE  ALT=\"x%2A%281%2Bsqrt%282%29%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F%281%2Bsqrt%282%29%29 ALIGN=MIDDLE  ALT=\"x=1%2F%281%2Bsqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28sqrt%282%29-1%29%2F%28%281%2Bsqrt%282%29%29%28sqrt%282%29-1%29%29 ALIGN=MIDDLE  ALT=\"x=%28sqrt%282%29-1%29%2F%28%281%2Bsqrt%282%29%29%28sqrt%282%29-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28sqrt%282%29-1%29%2F%28%28sqrt%282%29%29%5E2-1%5E2%29 ALIGN=MIDDLE  ALT=\"x=%28sqrt%282%29-1%29%2F%28%28sqrt%282%29%29%5E2-1%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28sqrt%282%29-1%29%2F%282-1%29 ALIGN=MIDDLE  ALT=\"x=%28sqrt%282%29-1%29%2F%282-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%282%29-1 ALIGN=MIDDLE  ALT=\"x=sqrt%282%29-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "USING AREA OF RED SQUARE MINUS AREA OF CORNER TRIANGLES:<BR>\n" );
document.write( "The area of the red square, in square feet is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A1=1 ALIGN=MIDDLE  ALT=\"1%2A1=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The corner triangles can be rearranged<BR>\n" );
document.write( "(putting all four right angles together)<BR>\n" );
document.write( "into a square of side <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%282%29-1 ALIGN=MIDDLE  ALT=\"x=sqrt%282%29-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with area<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%282%29-1%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%282%29-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the area of the octagon, in square feet, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-%28sqrt%282%29-1%29%5E2=1-%282-2sqrt%282%29%2B1%29=1-2%2B2sqrt%282%29-1=highlight%282sqrt%282%29-2=2%28sqrt%282%29-1%29=approximately0.8284%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "USING THE FORMULA FOR THE AREA OF A REGULAR POLYGON:<BR>\n" );
document.write( "The length of a side of the octagon, in feet, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%282%29-1 ALIGN=MIDDLE  ALT=\"x=sqrt%282%29-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perimeter=8%28sqrt%282%29-1%29 ALIGN=MIDDLE  ALT=\"perimeter=8%28sqrt%282%29-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The apothem of the octagon is<BR>\n" );
document.write( "the distance from the center of the octagon to one of its sides,<BR>\n" );
document.write( "it is also the distance from the center of the square to one of its sides,<BR>\n" );
document.write( "and that is half of a square side <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > foot.<BR>\n" );
document.write( "Substituting into the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=%281%2F2%29%2Aperimeter%2Aapothem ALIGN=MIDDLE  ALT=\"area=%281%2F2%29%2Aperimeter%2Aapothem\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=%281%2F2%29%2A8%28sqrt%282%29-1%29%2A%281%2F2%29=highlight%282%28sqrt%282%29-1%29=approximately0.8284%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );