document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1005912>Question 1005912</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Good day! Can you please help me with this problem?\r<BR>\n" );
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document.write( "The lateral edge of a regular hexagonal pyramid is two times the length of the base edge.  If the apothem of the base is 8 cm, find the altitude and volume of the cone inscribed in a pyramid.\r<BR>\n" );
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document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #622225 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52782)</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=622225\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "The lateral edge of a regular hexagonal pyramid is two times the length of the base edge. <BR>\n" );
document.write( "If the apothem of the base is 8 cm, find the altitude and volume of the cone inscribed in a pyramid. <BR>\n" );
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document.write( "Wikipedia <A  rel=nofollow HREF=https://en.wikipedia.org/wiki/Apothem>says</A>:\r\n" );
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document.write( "\"The apothem of a regular polygon is a line segment from the center to the midpoint of one of its sides. \r\n" );
document.write( "Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. \r\n" );
document.write( "The word \"apothem\" can also refer to the length of that line segment. Regular polygons are the only polygons \r\n" );
document.write( "that have apothems. Because of this, all the apothems in a polygon are be congruent\".\r\n" );
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document.write( "The <B>Figure</B> shows a regular hexagonal pyramid, &nbsp;an apothem of the regular hexagon in its base&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <BR>\n" );
document.write( "(OP), &nbsp;the pyramid's height &nbsp;(RO) &nbsp;and its slant height &nbsp;(RP).\r<BR>\n" );
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document.write( "The length of the apothem &nbsp;OP &nbsp;is given: &nbsp;|OP| = 8 cm.\r<BR>\n" );
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document.write( "We also know that the length of the lateral edge is twice the base edge length: &nbsp;|AR| = 2*|AB|.\r<BR>\n" );
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document.write( "The apothem &nbsp;OP &nbsp;is the height of the regular triangle &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA ALIGN=MIDDLE  ALT=\"DELTA\"   DISABLED_style= \"max-width:90%; height:auto;\" >OAB.\r<BR>\n" );
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document.write( "Therefore, &nbsp;|OP| = <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;\" >*|AB| = <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;\" >*|OA|. &nbsp;Hence, &nbsp;|OA| = |AB| = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"2%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >*|OP| = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"2%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >*8 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"16%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Since the length of the lateral edge is twice the base edge length, we have <BR>\n" );
document.write( "|AR| = 2*|AB| = 2*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"16%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now from the right-angled triangle &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA ALIGN=MIDDLE  ALT=\"DELTA\"   DISABLED_style= \"max-width:90%; height:auto;\" >AOR &nbsp;we have for the height of the pyramid\r<BR>\n" );
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document.write( "|OR| = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28abs%28AR%29%5E2+-+abs%28OA%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28abs%28AR%29%5E2+-+abs%28OA%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%282%2A16%2Fsqrt%283%29%29%5E2+-+%2816%2Fsqrt%283%29%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%282%2A16%2Fsqrt%283%29%29%5E2+-+%2816%2Fsqrt%283%29%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28+%284%2A16%5E2%29%2F3+-+%2816%5E2%29%2F3%29+%29 ALIGN=MIDDLE  ALT=\"sqrt%28+%284%2A16%5E2%29%2F3+-+%2816%5E2%29%2F3%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28+%283%2A16%5E2%29%2F3+%29 ALIGN=MIDDLE  ALT=\"sqrt%28+%283%2A16%5E2%29%2F3+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 16 cm.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;Regular hexagonal pyramid, <BR>\n" );
document.write( "its height and a slant height.\r<BR>\n" );
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document.write( "Thus for the inscribed cone you just know the radius of its base |OP| = 8 cm and its height |OR| = 16 cm. \r<BR>\n" );
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document.write( "It means that you can easily calculate the volume of the inscribed cone V = <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=pi%2Aabs%28OA%29%5E2%2Aabs%28OR%29 ALIGN=MIDDLE  ALT=\"pi%2Aabs%28OA%29%5E2%2Aabs%28OR%29\"   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=pi%2A8%5E2%2A16 ALIGN=MIDDLE  ALT=\"pi%2A8%5E2%2A16\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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