document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=472968>Question 472968</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you find the slant height and base edge of a regular square pyramid when the height is 6 and the lateral edge is 10? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #324355 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=324355\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You have a reguar square pyramkid.<BR>\n" );
document.write( "The base is square.<BR>\n" );
document.write( "The height is 6.<BR>\n" );
document.write( "The lateral edge is 10.<BR>\n" );
document.write( "Here's a reference that tells you all sorts of things related to pyramids.<BR>\n" );
document.write( "One of those things describes what the lateral edge is and tells you how to calculate it given the radius of the base and the height.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.vitutor.com/geometry/solid/pyramid_math.html\" target = \"_blank\">http://www.vitutor.com/geometry/solid/pyramid_math.html</a><BR>\n" );
document.write( "In your problem, you are given the height and the lateral edge and you are asked to find the slant height.<BR>\n" );
document.write( "That's slightly different.<BR>\n" );
document.write( "The slant height would be the altitude of any of the lateral faces.<BR>\n" );
document.write( "Lateral face and slant face are the same as far as I can tell.<BR>\n" );
document.write( "Check the reference for a description of what a lateral face is.<BR>\n" );
document.write( "To find the slant height, you need to find the apothem of the base.<BR>\n" );
document.write( "To find the apothem of the base, you need to find the radius of the base.<BR>\n" );
document.write( "the picture below shows you what your pyramid will look like.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/problems/pyramid_1.jpg\" alt = \"***** picture not found *****\"/ ><BR>\n" );
document.write( "AE is a lateral edge.<BR>\n" );
document.write( "AB is the height of the pyramid.<BR>\n" );
document.write( "BE is radius of the base of the pyramid.  If you were to circumscribe a circle around the base of the pyramid, BE would be the radius of that circle.<BR>\n" );
document.write( "BG is the apothem of the base of the pyramid.  If you were to inscribe a circle within the base of the pyramid, BG would be the radius of that circle.<BR>\n" );
document.write( "Angle x is the angle that you need to find the radius of the base.<BR>\n" );
document.write( "Angle x is angle AEB.<BR>\n" );
document.write( "The sine of angle x is equal to opposite / hypotenuse which is equal to AB / AE which is equal to 6/10 which is equal to .6<BR>\n" );
document.write( "Angle x is equal to arcsine of .6 which is equal to 36.86989765 degrees.<BR>\n" );
document.write( "Cosine of angle x is equal to adjacent / hypotenuse which is equal to BE / AE which is equal to BE / 10.<BR>\n" );
document.write( "Multiply both sides of this equation by 10 and you get 10 * cosine of angle x is equal to BE.<BR>\n" );
document.write( "This makes BE equal to 8 as shown in the diagram.<BR>\n" );
document.write( "You now have enough information to find BG which is the apothem of the base.<BR>\n" );
document.write( "In order to find the apothem of the base, you need to know what angle y is.<BR>\n" );
document.write( "angle y is angle BEG.<BR>\n" );
document.write( "Since the base is a square, angle y is equal to 45 degrees.<BR>\n" );
document.write( "The sine of angle y is equal to opposite / hypotenuse which is equal to BG / BE which is equal to BG / 8.<BR>\n" );
document.write( "Multiply both sides of this equation by 8 to get:<BR>\n" );
document.write( "8 * sine (45) = BG.<BR>\n" );
document.write( "This makes BG equal to 5.65685425.<BR>\n" );
document.write( "You now have enough information to find the slant height.<BR>\n" );
document.write( "In order to do that, you need to find angle z.<BR>\n" );
document.write( "Angle z is angle AGB.<BR>\n" );
document.write( "The tangent of angle z is equal to opposite / adjacent which is equal to AB / BG which is equal to 6 / 5.65685425 which is equal to 1.060660172.<BR>\n" );
document.write( "The arc tangent of 1.060660172 is equal to 46.68614334.<BR>\n" );
document.write( "Angle y is equal to 46.68614334.<BR>\n" );
document.write( "You can now find the slant height.<BR>\n" );
document.write( "The sine of angle z is equal to opposite / hypotenuse which is equal to AB / AG which is equal to 6 / AG.<BR>\n" );
document.write( "Multiply both sides of this equation by AG and divide both sides of this equation by sine (z) to get:<BR>\n" );
document.write( "AG = 6 / sine(z)<BR>\n" );
document.write( "Since z equals 46.68614334 degrees, this equation becomes:<BR>\n" );
document.write( "AG = 6 / sine (46.68614334) which is equal to 6 / .727606875 which is equal to 8.246211251.<BR>\n" );
document.write( "AG is equal to 8.246211251 which is equal to the slant height of your pyramid.<BR>\n" );
document.write( "The following picture shows you the details of each section of the pyramid as we progress from the start to the solution.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/problems/pyramid_2.jpg\" alt = \"***** picture not found *****\"/ ><BR>\n" );
document.write( "Step 1 is where we find the radius of the base.<BR>\n" );
document.write( "Step 2 is where we find the apothem of the base.<BR>\n" );
document.write( "Stepo 3 is where we find the slant height of the pyramid.\r<BR>\n" );
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