document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1113151>Question 1113151</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a circle with a diameter of 12 inches, a regular five-pointed star is inscribed. What is the area of that part not covered by the star? Please explain. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #728182 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=728182\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The regular five-pointed star,<BR>\n" );
document.write( "and the circle with radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12inch%2F2=6inch%29 ALIGN=MIDDLE  ALT=\"12inch%2F2=6inch%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where it is inscribed,<BR>\n" );
document.write( "would look like this:<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28330%2C330%2C-6.6%2C6.6%2C-6.6%2C6.6%2Cgrid%280%29%2C%0D%0Ared%28circle%280%2C0%2C6%29%29%2Cline%280%2C6%2C3.527%2C-4.854%29%2C%0D%0Aline%280%2C6%2C-3.527%2C-4.854%29%2Cline%28-3.527%2C-4.854%2C5.706%2C1.854%29%2C%0D%0Aline%28-5.706%2C1.854%2C5.706%2C1.854%29%2Cline%28-5.706%2C1.854%2C3.527%2C-4.854%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > We see that the star is a pentagon plus 5 isosceles triangles.<BR>\n" );
document.write( "If you were given a formula to calculate areas of regular n-pointed stars inscribed in circles,<BR>\n" );
document.write( "by all means use those formulas.<BR>\n" );
document.write( "I think it is better not to use a formula if you do not have the formulas,<BR>\n" );
document.write( "and are not able to deduce one quickly on the spot.<BR>\n" );
document.write( "I can deduce one, but I prefer not to let formulas interfere with visualizing and understanding, so I will show a different approach.<BR>\n" );
document.write( " <BR>\n" );
document.write( "We can figure out some angles:<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28330%2C330%2C-6.6%2C6.6%2C-6.6%2C6.6%2Cgrid%280%29%2C%0D%0Ared%28circle%280%2C0%2C6%29%29%2Cline%280%2C6%2C3.527%2C-4.854%29%2Clocate%285.8%2C1.9%2CP%29%2C%0D%0Aline%280%2C6%2C-3.527%2C-4.854%29%2Cline%28-3.527%2C-4.854%2C5.706%2C1.854%29%2C%0D%0Aline%28-5.706%2C1.854%2C5.706%2C1.854%29%2Cline%28-5.706%2C1.854%2C3.527%2C-4.854%29%2C%0D%0Ared%28arrow%280%2C0%2C6.562%2C2.132%29%29%2Cred%28arc%280%2C0%2C4.4%2C4.4%2C-18%2C0%29%29%2C%0D%0Alocate%282.2%2C1%2Cred%2890%5Eo-72%5Eo=18%5Eo%29%29%2Cred%28arc%280%2C0%2C8%2C8%2C-90%2C-18%29%29%2C%0D%0Alocate%282.8%2C3.5%2Cred%2872%5Eo%29%29%2Cgreen%28arrow%280%2C0%2C3.879%2C5.34%29%29%2C%0D%0Agreen%28arc%280%2C0%2C2.2%2C2.2%2C-90%2C-54%29%29%2Clocate%280.1%2C2%2Cgreen%2836%5Eo%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "The angle between the top point of the star and point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%5BP%5D%2Cy%5BP%5D%29 ALIGN=MIDDLE  ALT=\"P%28x%5BP%5D%2Cy%5BP%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the rightmost point of the 5-pointed star)<BR>\n" );
document.write( "measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo%2F5=72%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo%2F5=72%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the complementary angle,<BR>\n" );
document.write( "the one between the positive x-axis and the red ray going through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90%5Eo-72%5Eo=18%5Eo ALIGN=MIDDLE  ALT=\"90%5Eo-72%5Eo=18%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With that we can calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BP%5D=6%2Asin%2818%5Eo%29 ALIGN=MIDDLE  ALT=\"y%5BP%5D=6%2Asin%2818%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F0.309017=about1.854 ALIGN=MIDDLE  ALT=\"6%2F0.309017=about1.854\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is (in inches) the apothem of the pentagon, while the height (in inches) of the isosceles triangles is about<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=height=6-1.854=4.146 ALIGN=MIDDLE  ALT=\"height=6-1.854=4.146\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Now we only need the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perimeter ALIGN=MIDDLE  ALT=\"perimeter\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the pentagon,<BR>\n" );
document.write( "and then we would be able to calculate the area of the pentagon as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perimeter%2Aapothem%2F2 ALIGN=MIDDLE  ALT=\"perimeter%2Aapothem%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the area of the 5 triangles that make the points of the star as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perimeter%2Aheight%2F2 ALIGN=MIDDLE  ALT=\"perimeter%2Aheight%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There must be other formulas that we could look up,<BR>\n" );
document.write( "but knowing that the angle marked with a green arc<BR>\n" );
document.write( "(half of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72%5Eo ALIGN=MIDDLE  ALT=\"72%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > central angles of the pentagon),<BR>\n" );
document.write( "measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72%5Eo%2F2=36%5Eo ALIGN=MIDDLE  ALT=\"72%5Eo%2F2=36%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "we can calculate the length (in inches) of half a side of the pentagon as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=apothem%2Atan%2836%5Eo%29=about1.854%2A0.891=about1.652 ALIGN=MIDDLE  ALT=\"apothem%2Atan%2836%5Eo%29=about1.854%2A0.891=about1.652\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That makes the perimeter of the pentagon approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2A1.652inches=16.520inches ALIGN=MIDDLE  ALT=\"10%2A1.652inches=16.520inches\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The area of the pentagon (in square inches) is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16.520%2A1.854%2A2=13.514 ALIGN=MIDDLE  ALT=\"16.520%2A1.854%2A2=13.514\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The area (in square inches) of the five triangles forming the tips of the star is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16.52%2A4.146%2A2=34.246 ALIGN=MIDDLE  ALT=\"16.52%2A4.146%2A2=34.246\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The area (in square inches) of 5-pointed star is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13.514%2B34.246=490.561 ALIGN=MIDDLE  ALT=\"13.514%2B34.246=490.561\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "The area (in square inches) of the circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A4%5E2=36pi=about113.097 ALIGN=MIDDLE  ALT=\"pi%2A4%5E2=36pi=about113.097\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the area of that part not covered by the star is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=113.097in%5E2-49.561in%5E2=highlight%2863.537in%5E2%29 ALIGN=MIDDLE  ALT=\"113.097in%5E2-49.561in%5E2=highlight%2863.537in%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );