document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=179127>Question 179127</A>: <I><SPAN STYLE=\"word-break: break-all;\"> why does the volume of a sphere does have 4 in it's formula?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #134092 by </B><A HREF=/tutors/aboutme.mpl?userid=Mathtut><B>Mathtut(3670)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Mathtut><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=134092\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> that is a pretty complicated question:<BR>\n" );
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document.write( "the derivation of the Volume of a sphere involves Calculus.<BR>\n" );
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document.write( "The answer is not easy to illustrate. here is a partial explanation<BR>\n" );
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document.write( "Let's start with the surface area.  Take a sphere of radius R, and <BR>\n" );
document.write( "imagine constructing a cylindrical box with radius R and height 2R.  <BR>\n" );
document.write( "You can see that the sphere will fit snugly inside this box.\r<BR>\n" );
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document.write( "Archimedes, the Greek mathematician, proved a surprising fact: the <BR>\n" );
document.write( "surface area of the sphere is exactly the same as the lateral surface <BR>\n" );
document.write( "area of the cylinder (that is, the surface area not including the two <BR>\n" );
document.write( "circular ends).  You can calculate the lateral surface area of the <BR>\n" );
document.write( "cylinder and you will see that it is 4*pi*R^2.  The following item in <BR>\n" );
document.write( "the Dr. Math Archives describes what Archimedes did to prove this <BR>\n" );
document.write( "result:\r<BR>\n" );
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document.write( "  In brief, you can imagine drawing a tiny triangle on <BR>\n" );
document.write( "the surface of the sphere and connecting its corners to the center of <BR>\n" );
document.write( "the sphere.  You have made a very narrow pyramid.  The volume of a <BR>\n" );
document.write( "pyramid is 1/3 times the area of the base times the height.  Thus the <BR>\n" );
document.write( "volume of this pyramid is 1/3 times the radius of the sphere, times <BR>\n" );
document.write( "the area of that little triangle.\r<BR>\n" );
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document.write( "Now, imagine that you cover the sphere with tiny triangles, and thus <BR>\n" );
document.write( "cut the sphere into millions of narrow pyramids.  The total volume of <BR>\n" );
document.write( "the pyramids is 1/3 times the radius of the sphere times the sum of <BR>\n" );
document.write( "the areas of the tiny triangles.  In other words, the volume of the <BR>\n" );
document.write( "sphere is 1/3 times R times the surface area of the sphere!\r<BR>\n" );
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document.write( "  V = (1/3)R * 4*pi*R^2<BR>\n" );
document.write( "    = (4/3)pi*R^3<BR>\n" );
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