document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=409794>Question 409794</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the 2 column proof for theorem 11.8 (circumference of a circle)?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #288345 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=288345\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Proving the circumference of a circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Api%2Ar ALIGN=MIDDLE  ALT=\"2%2Api%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a little tricky using a two-column proof. It's also difficult to construct a proof of the circumference without using calculus.\r<BR>\n" );
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document.write( "The best way is to consider a regular n-sided polygon and its perimeter. Suppose the distance from the center of an n-sided polygon to one of its vertices is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > (where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > is constant). We can construct <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > isosceles triangles, each one looking something like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28200%2C+200%2C+0%2C+6%2C+0%2C+10%2C%0D%0A%0D%0A%0D%0Atriangle%282%2C+2%2C+4%2C+2%2C+3%2C+8%29%2C%0D%0Alocate%282%2C+5%2C+r%29%2C%0D%0Alocate%282.9%2C+7%2C+theta%29%0D%0A%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+2%2Api%2Fn ALIGN=MIDDLE  ALT=\"theta+=+2%2Api%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Hence, the base of the isosceles triangle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2r%2Asin%28theta%2F2%29+=+2r%2Asin%28pi%2Fn%29 ALIGN=MIDDLE  ALT=\"2r%2Asin%28theta%2F2%29+=+2r%2Asin%28pi%2Fn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (applying the definition of sine). We have n of these segments so the perimeter of the n-gon is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2rn%2Asin%28pi%2Fn%29 ALIGN=MIDDLE  ALT=\"2rn%2Asin%28pi%2Fn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Taking the limit as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes to infinity,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+lim%28n-%3Einfinity%2C+2rn%2Asin%28pi%2Fn%29%29 ALIGN=MIDDLE  ALT=\"C+=+lim%28n-%3Einfinity%2C+2rn%2Asin%28pi%2Fn%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+1%2Fn ALIGN=MIDDLE  ALT=\"u+=+1%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This limit is equivalent to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+lim%28u-%3E0%2C+2r%2Asin%28u%2Api%29%2Fu%29 ALIGN=MIDDLE  ALT=\"C+=+lim%28u-%3E0%2C+2r%2Asin%28u%2Api%29%2Fu%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Applying L'Hopital's rule this is equivalent to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+lim%28u-%3E0%2C+2r%2Api%2Acos%28u%2Api%29%29+=+2%2Api%2Ar ALIGN=MIDDLE  ALT=\"C+=+lim%28u-%3E0%2C+2r%2Api%2Acos%28u%2Api%29%29+=+2%2Api%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is the circumference of the circle. </SPAN>\n" );
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