document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=536015>Question 536015</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two circles both with radius 4, the centre of each circle lies on the circumference of the other, what is the exact area which is common to both circles? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #352134 by </B><A HREF=/tutors/aboutme.mpl?userid=htmentor><B>htmentor(1343)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=htmentor><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=htmentor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=352134\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Two circles both with radius 4, the centre of each circle lies on the circumference of the other, what is the exact area which is common to both circles?<BR>\n" );
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document.write( "The area of a circular sector = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2%2Atheta+%2F2 ALIGN=MIDDLE  ALT=\"r%5E2%2Atheta+%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the sector angle and <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 the radius<BR>\n" );
document.write( "The sector consists of a circular segment plus a triangular portion <BR>\n" );
document.write( "[think of an ice cream cone where the cone is the triangular portion and the segment is the ice cream]<BR>\n" );
document.write( "The height of the triangle = 2 [half the width of the overlapping portion]<BR>\n" );
document.write( "The base of the triangle = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%284%5E2+-+2%5E2%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%284%5E2+-+2%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the area of the triangle = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%2812%29+=+4%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%2812%29+=+4%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Aarccos%282%2F4%29 ALIGN=MIDDLE  ALT=\"2%2Aarccos%282%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Api%2F3 ALIGN=MIDDLE  ALT=\"2%2Api%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area of the segment = the area of the sector - the area of the triangle<BR>\n" );
document.write( "Area(segment) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%2Api%2F3+-+4%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"16%2Api%2F3+-+4%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There are two segments which make up the overlapping area, so the area common to both cirles is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2A%284%2Api%2F3+-+sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"8%2A%284%2Api%2F3+-+sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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