document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=904401>Question 904401</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a regular polygon of 24 sides is inscribed in a circle of radius 1 cm , then find the length of each side of the polygon  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #548661 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=548661\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula is s = 2*r*sin(180/n)<BR>\n" );
document.write( "s = length of a side<BR>\n" );
document.write( "r = radius of the circle<BR>\n" );
document.write( "n = number of sides of the polygon<BR>\n" );
document.write( "in your problem this becomes:<BR>\n" );
document.write( "s = 2*1*sin(180/24) which becomes s = 2*1*sin(7.5) which becomes s = .2610523844.\r<BR>\n" );
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document.write( "the formula is derived from the fact that a regular polygon of 24 sides forms 24 isosceles triangle with side legs of 1 inch each and a central angle, or vertex angle of the isosceles triangle, of 15 degrees.\r<BR>\n" );
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document.write( "drop an altitude from the vertex to the base of the isosceles triangle and that forms two right triangles that are congruent to each other.\r<BR>\n" );
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document.write( "each of these right triangles has a central angle of 7.5 degrees.\r<BR>\n" );
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document.write( "the sine of the central angle of 7.5 degrees is equal to the side opposite the central angle divided by the hypotenuse which give you the formula of sine(7.5) = x/1\r<BR>\n" );
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document.write( "solve for x and you get x = 1 * sin(7.5)\r<BR>\n" );
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document.write( "you want 2x because x is only 1/2 the length of a side of the polygon, so the formula becomes s = 2*sin(7.5)\r<BR>\n" );
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document.write( "that gets you s = .26105.....\r<BR>\n" );
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document.write( "2*sin(7.5) is equivalent to the generic formula of 2*r*sin(180/n) where r is equal to 1 and n is equal to 24.\r<BR>\n" );
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document.write( "the formula becomes s = 2*1*sin(180/24) which becomes s = 2*sin(7.5) as we saw above.\r<BR>\n" );
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