document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1061891>Question 1061891</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given six congruent circles drawn internally tangent to a circle of radius 18; each smaller circle is also tangent to each of its adjacent circles. Find the shaded area between the large circle and the six smaller circles. \r<BR>\n" );
document.write( "\n" );
document.write( "Image is in the link: http://prntscr.com/dkmacg </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #676718 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=676718\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-3.3%2C3.3%2C-3.3%2C3.3%2C%0D%0Acircle%280%2C0%2C3%29%2C%0D%0Acircle%28-2%2C0%2C1%29%2Ccircle%282%2C0%2C1%29%2C%0D%0Acircle%281%2C1.732%2C1%29%2Ccircle%28-1%2C1.732%2C1%29%2C%0D%0Acircle%281%2C-1.732%2C1%29%2Ccircle%28-1%2C-1.732%2C1%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > + another small circle = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-3.3%2C3.3%2C-3.3%2C3.3%2C%0D%0Acircle%280%2C0%2C3%29%2Cgreen%28circle%280%2C0%2C1%29%29%2C%0D%0Acircle%28-2%2C0%2C1%29%2Ccircle%282%2C0%2C1%29%2C%0D%0Acircle%281%2C1.732%2C1%29%2Ccircle%28-1%2C1.732%2C1%29%2C%0D%0Acircle%281%2C-1.732%2C1%29%2Ccircle%28-1%2C-1.732%2C1%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The smaller circles' diameter is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the diameter of the smaller circle. <BR>\n" );
document.write( "Radius of the larger circle = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Radius of each smaller circle = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2F3=6 ALIGN=MIDDLE  ALT=\"18%2F3=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Area of each smaller circle = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=pi%2A6%5E2=36pi ALIGN=MIDDLE  ALT=\"S=pi%2A6%5E2=36pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the larger circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > times as wide,<BR>\n" );
document.write( "its area, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L ALIGN=MIDDLE  ALT=\"L\"   DISABLED_style= \"max-width:90%; height:auto;\" > , is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E2=9 ALIGN=MIDDLE  ALT=\"3%5E2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > times larger:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=9S ALIGN=MIDDLE  ALT=\"L=9S\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "The area between the large circle and the six smaller circles is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L-6S=9S-6S=3S=3%2836pi%29=highlight%28108pi%29 ALIGN=MIDDLE  ALT=\"L-6S=9S-6S=3S=3%2836pi%29=highlight%28108pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "You may be familiar with this arrangement of 6 circles around a central one,<BR>\n" );
document.write( "because you can form it with 7 pennies (or seven coins, or right cylindrical objects of the same shape).<BR>\n" );
document.write( "Do you also see it in \"pilings od same sized cylindrical objects,<BR>\n" );
document.write( "such as a flatbed truck cargo of metal pipes or plastic tubes.<BR>\n" );
document.write( "Do you want a geometry proof?<BR>\n" );
document.write( "Joining the centers of adjacent circles with line segments will give you a regular hexagon.<BR>\n" );
document.write( "Adding 3 diameters of the outside circle passing through the hexagon vertices,<BR>\n" );
document.write( "you have  the hexagon split into 6 triangles.<BR>\n" );
document.write( "The central angles measure <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo%2F6=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo%2F6=360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the triangle sides flanking those angles are congruent,<BR>\n" );
document.write( "meaning that the triangles are isosceles, with a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60%5Eo ALIGN=MIDDLE  ALT=\"60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > vertex angle.<BR>\n" );
document.write( "So, in the triangles, the base angles (adjacent to the hexagon sides)<BR>\n" );
document.write( "must be congruent, and must measure<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28180%5Eo-60%5Eo%29%2F2=120%5Eo%2F2=60%5Eo ALIGN=MIDDLE  ALT=\"%28180%5Eo-60%5Eo%29%2F2=120%5Eo%2F2=60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, those isosceles triangles (and all isoceles triangles with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60%5Eo ALIGN=MIDDLE  ALT=\"60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > vertex angles) are equilateral triangles.<BR>\n" );
document.write( "All sides of those triangles are congruent,<BR>\n" );
document.write( "all measuring 2 times the radius of the smaller circles.<BR>\n" );
document.write( "That means that there is definitely just enough room in the center for another smaller circle. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );