document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1014348>Question 1014348</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A hexagon is cut from a square of side 6 inches.<BR>\n" );
document.write( "a. What is the apothem of the hexagon?<BR>\n" );
document.write( "b. What is the area of the hexagon?<BR>\n" );
document.write( "c. Find the perimeter of the hexagon. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #630718 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=630718\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> What kind of a hexagon? It must be a regular hexagon, because that is the kind of hexagon that has an apothem.<BR>\n" );
document.write( "How is the regular hexagon expected to be cut out of that square? With two sides parallel to sides of the square,<BR>\n" );
document.write( "like this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.2%2C1.2%2C-1.2%2C1.2%2C%0D%0Arectangle%28-1%2C-1%2C1%2C1%29%2Cgreen%28circle%280%2C0%2C1%29%29%2C%0D%0Aline%281%2C0%2C0.5%2C0.866%29%2Cline%280.5%2C0.866%2C-0.5%2C0.866%29%2C%0D%0Aline%28-0.5%2C0.866%2C-1%2C0%29%2Cline%28-1%2C0%2C-0.5%2C-0.866%29%2C%0D%0Aline%28-0.5%2C-0.866%2C0.5%2C-0.866%29%2Cline%280.5%2C-0.866%2C1%2C0%29%2C%0D%0Aarrow%28-0.5%2C0.866%2C0%2C0.866%29%2Carrow%28-0.5%2C-0.866%2C0%2C-0.866%29%2C%0D%0Aarrow%28-0.5%2C1%2C0%2C1%29%2Carrow%28-0.5%2C-1%2C0%2C-1%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "I assume that is the basis of the answer expected.<BR>\n" );
document.write( "That hexagon is inscribed in the circle inscribed in the square.<BR>\n" );
document.write( "It is made of six equilateral triangles whose side lengths are half the length of the square side.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.2%2C1.2%2C-1.2%2C1.2%2C%0D%0Arectangle%28-1%2C-1%2C1%2C1%29%2Cgreen%28circle%280%2C0%2C1%29%29%2C%0D%0Aline%281%2C0%2C0.5%2C0.866%29%2Cline%280.5%2C0.866%2C-0.5%2C0.866%29%2C%0D%0Aline%28-0.5%2C0.866%2C-1%2C0%29%2Cline%28-1%2C0%2C-0.5%2C-0.866%29%2C%0D%0Aline%28-0.5%2C-0.866%2C0.5%2C-0.866%29%2Cline%280.5%2C-0.866%2C1%2C0%29%2C%0D%0Aline%28-1%2C0%2C1%2C0%29%2Cline%280.5%2C0.866%2C-0.5%2C-0.866%29%2C%0D%0Aline%28-0.5%2C0.866%2C0.5%2C-0.866%29%2Clocate%28-0.1%2C-1%2C6in%29%2C%0D%0Alocate%28-0.6%2C0%2C3in%29%2Clocate%280.4%2C0%2C3in%29%2C%0D%0Aarrow%28-0.12%2C-1.06%2C-1%2C-1.06%29%2Carrow%280.12%2C-1.06%2C1%2C-1.06%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The apothem is the height of one of those triangles: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.1%2C1.1%2C-0.2%2C2%2C%0D%0Agreen%28triangle%28-1%2C0%2C0%2C0%2C0%2C1.732%29%29%2C%0D%0Agreen%28rectangle%280%2C0%2C0.1%2C0.1%29%29%2C%0D%0Atriangle%28-1%2C0%2C1%2C0%2C0%2C1.732%29%2C%0D%0Alocate%28-0.55%2C0.12%2C1.5in%29%2Clocate%280.45%2C0.12%2C1.5in%29%2C%0D%0Alocate%28-0.5%2C0.87%2C3in%29%2Clocate%280.35%2C0.87%2C3in%29%2C%0D%0Alocate%280.02%2C0.6%2Cgreen%28apothem%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "According to the Pythagorean theorem, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=apothem%5E2%2B%281.5in%29%5E2=%283in%29%5E2 ALIGN=MIDDLE  ALT=\"apothem%5E2%2B%281.5in%29%5E2=%283in%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=apothem%5E2=%283in%29%5E2-%281.5in%29%5E2 ALIGN=MIDDLE  ALT=\"apothem%5E2=%283in%29%5E2-%281.5in%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=apothem%5E2=9in%5E2-2.25in%5E2 ALIGN=MIDDLE  ALT=\"apothem%5E2=9in%5E2-2.25in%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=apothem%5E2=6.75in%5E2 ALIGN=MIDDLE  ALT=\"apothem%5E2=6.75in%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=apothem=1.5sqrt%283%29 ALIGN=MIDDLE  ALT=\"apothem=1.5sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=in=about2.598in ALIGN=MIDDLE  ALT=\"in=about2.598in\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded).<BR>\n" );
document.write( "Since the area of a regular polygon can be calculated as<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;\" > ,<BR>\n" );
document.write( "and the perimeter of a hexagon with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3in ALIGN=MIDDLE  ALT=\"3in\"   DISABLED_style= \"max-width:90%; height:auto;\" > sides is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%283in%29=18in ALIGN=MIDDLE  ALT=\"6%283in%29=18in\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=%2818in%29%28about2.598in%29%2F2=about23.38in%5E2 ALIGN=MIDDLE  ALT=\"area=%2818in%29%28about2.598in%29%2F2=about23.38in%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "A more creative person could cut out a slightly larger hexagon,<BR>\n" );
document.write( "like this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28311%2C311%2C-1.2%2C1.2%2C-1.2%2C1.2%2C%0D%0Arectangle%28-.966%2C-.966%2C.966%2C.966%29%2Cgreen%28circle%280%2C0%2C1%29%29%2C%0D%0Aline%280.966%2C0.259%2C0.259%2C0.966%29%2Cline%280.259%2C0.966%2C-0.707%2C0.707%29%2C%0D%0Aline%28-0.707%2C0.707%2C-0.966%2C-0.259%29%2Cline%28-0.966%2C-0.259%2C-0.259%2C-0.966%29%2C%0D%0Aline%28-0.259%2C-0.966%2C0.707%2C-0.707%29%2Cline%280.707%2C-0.707%2C0.966%2C0.259%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The side of that hexagon is about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.035 ALIGN=MIDDLE  ALT=\"1.035\"   DISABLED_style= \"max-width:90%; height:auto;\" > times larger, and so is the apothem.<BR>\n" );
document.write( "The area of that hexagon is about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.035%5E2=about1.07 ALIGN=MIDDLE  ALT=\"1.035%5E2=about1.07\"   DISABLED_style= \"max-width:90%; height:auto;\" > times larger. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );