document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1036471>Question 1036471</A>: <I><SPAN STYLE=\"word-break: break-all;\"> All the edges of a regular triangular pyramid are x units long. Find the volume of the pyramid in terms of x. In the end, a formula for volume should result. Show the steps taken to get to this formula. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #651181 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52946)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=651181\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "All the edges of a regular triangular pyramid are x units long. Find the volume of the pyramid in terms of x. <BR>\n" );
document.write( "In the end, a formula for volume should result. Show the steps taken to get to this formula.<BR>\n" );
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document.write( "<B>Solution</B>\r<BR>\n" );
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document.write( "The base area of the given tetrahedron is the area of the equilateral triangle with&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\r<BR>\n" );
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document.write( "the side measure x. &nbsp;So, &nbsp;the base area is equal to &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bbase%5D ALIGN=MIDDLE  ALT=\"S%5Bbase%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2Asqrt%283%29%29%2F4 ALIGN=MIDDLE  ALT=\"%28x%5E2%2Asqrt%283%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Next, &nbsp;let us find the measure of the height of the given tetrahedron &nbsp;(<B>Figure 1b</B>).\r<BR>\n" );
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document.write( "The height &nbsp;<B>OP</B>&nbsp; of the given tetrahedron drops to the center &nbsp;<B>O</B>&nbsp; of its base, &nbsp;which \r<BR>\n" );
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document.write( "is the intersection point of the base altitudes, &nbsp;medians and angle bisectors. \r<BR>\n" );
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document.write( "It is well known fact of &nbsp;<B>Planimetry</B>&nbsp; that the intersection point of medians of a \r<BR>\n" );
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document.write( "triangle divides them in proportion &nbsp;2:1&nbsp; counting from the vertices &nbsp;(see the lesson \r<BR>\n" );
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document.write( "<A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Triangles/Medians-of-a-triangle-are-concurrent.lesson>Medians of a triangle are concurrent</A>&nbsp; in this site). \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+250%2C+225%2C++-2.5%2C+4.5%2C+-0.8%2C+8.2%2C+%0D%0A%0D%0A++++++++++++line+%28++0.0%2C+0.0%2C++4.0%2C++0.0%29%2C%0D%0A++++++++++++line+%28++0.0%2C+0.0%2C+-2.0%2C++1.5%29%2C%0D%0A++++++green%28line+%28+-2.0%2C+++1.5%2C++4.0%2C++0.0%29%29%2C%0D%0A%0D%0A++++++++++++line+%28++0.667%2C+7.0%2C++0.0%2C++0.0%29%2C%0D%0A++++++++++++line+%28++0.667%2C+7.0%2C++4.0%2C++0.0%29%2C%0D%0A%0D%0A++++++++++++line+%28++0.667%2C+7.0%2C+-2.0%2C++1.5%29%2C%0D%0A%0D%0A++++++++++++locate%28+1.4%2C+-0.1%2C+x%29%2C%0D%0A++++++++++++locate%28-1.7%2C++0.9%2C+x%29%2C%0D%0A++++++++++++locate%28+1.2%2C++1.3%2C+x%29%2C%0D%0A%0D%0A++++++++++++locate%28+0.4%2C++3.7%2C+x%29%2C%0D%0A++++++++++++locate%28+2.5%2C++3.9%2C+x%29%2C%0D%0A++++++++++++locate%28-1.3%2C++4.4%2C+x%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure 1a</B>.  \r<BR>\n" );
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document.write( "Thus the segment &nbsp;<B>OA</B>&nbsp; in &nbsp;<B>Figure 1b</B>&nbsp; has the length of two third of the altitude \r<BR>\n" );
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document.write( "of the base triangle, &nbsp;i.e. &nbsp;|<B>OA</B>| = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F3 ALIGN=MIDDLE  ALT=\"2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Asqrt%283%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28x%2Asqrt%283%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Asqrt%283%29%29%2F3 ALIGN=MIDDLE  ALT=\"%28x%2Asqrt%283%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now, &nbsp;the height of the pyramid is &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28abs%28AP%29%5E2+-+abs%28OP%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28abs%28AP%29%5E2+-+abs%28OP%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2+-+%28x%2Asqrt%283%29%2F3%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2+-+%28x%2Asqrt%283%29%2F3%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Asqrt%281+-+1%2F3%29 ALIGN=MIDDLE  ALT=\"x%2Asqrt%281+-+1%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Asqrt%282%29%29%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"%28x%2Asqrt%282%29%29%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; \r<BR>\n" );
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document.write( "and the volume of our tetrahedron is &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <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;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bbase%5D ALIGN=MIDDLE  ALT=\"S%5Bbase%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <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;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2Asqrt%283%29%29%2F4 ALIGN=MIDDLE  ALT=\"%28x%5E2%2Asqrt%283%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Asqrt%282%29%29%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"%28x%2Asqrt%282%29%29%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3%2Asqrt%282%29%29%2F12 ALIGN=MIDDLE  ALT=\"%28x%5E3%2Asqrt%282%29%29%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<B>Answer</B>. &nbsp;The volume of the regular tetrahedron with the edge length x is &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3%2Asqrt%282%29%29%2F12 ALIGN=MIDDLE  ALT=\"%28x%5E3%2Asqrt%282%29%29%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure 1b</B>. \r<BR>\n" );
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