Lesson OVERVIEW of LESSONS on Volume of PYRAMIDS
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<H2>OVERVIEW of LESSONS on Volume of Pyramids</H2>For your convenience, this file contains - the list of my lessons on volume of pyramids in this site, - the formula for calculating the volume of pyramids, and - the list of relevant solved problems. <H3>Examples of pyramids</H3>Figures <B>1a</B> - <B>1e</B> present the examples of pyramids. <TABLE> <TR> <TD> {{{drawing( 200, 225, -3.0, 3.5, -0.5, 3.8, line ( 0.0, 0.0, 3.0, 0.0), line ( 0.0, 0.0, -2.0, 0.8), green(line ( -2.0, 0.8, 1.0, 0.8)), green(line ( 1.0, 0.8, 3.0, 0.0)), line ( 0.0, 0.0, 0.5, 3.4), line ( 3.0, 0.0, 0.5, 3.4), green(line ( 1.0, 0.8, 0.5, 3.4)), line ( -2.0, 0.8, 0.5, 3.4) )}}} <B>Figure 1a</B>. Rectangular pyramid </TD> <TD> {{{drawing( 250, 225, -5.0, 5.0, -0.5, 5.0, line ( 0.5, 0.0, 4.5, 0.0), line ( 0.5, 0.0, -4.0, 1.5), line ( 0.5, 0.0, -0.5, 4.5), line ( 4.5, 0.0, -0.5, 4.5), line ( -4.0, 1.5, -0.5, 4.5), green(line ( -0.5, 4.5, -0.5, 1.5)), green(line ( -4.0, 1.5, -0.5, 1.5)), green(line ( -0.5, 1.5, 4.5, 0.0)) )}}} <B>Figure 1b</B>. Rectangular pyramid </TD> <TD> {{{drawing( 225, 225, -4.0, 5.0, -0.5, 7.5, line ( -0.8, 0.0, 4.0, 0.0), line ( -0.8, 0.0, -3.0, 1.5), green(line ( -3.0, 1.5, 4.0, 0.0)), line ( -0.8, 0.0, 0.0, 7.0), line ( 0.0, 7.0, 4.0, 0.0), line ( 0.0, 7.0, -3.0, 1.5) )}}} <B>Figure 1c</B>. Triangular pyramid </TD> <TD> {{{drawing( 225, 225, -4.0, 5.0, -0.5, 8.5, line ( -3.0, 1.5, 4.0, 0.0), green(line ( -3.0, 1.5, 2.0, 2.0)), green(line ( 2.0, 2.0, 4.0, 0.0)), line ( -3.0, 1.5, 2.0, 8.0), line ( 4.0, 0.0, 2.0, 8.0), green(line ( 2.0, 2.0, 2.0, 8.0)) )}}} <B>Figure 1d</B>. Triangular pyramid </TD> <TD> {{{drawing( 238, 232, -2.5, 7.0, -0.5, 8.6, line ( 0.0, 0.0, 4.0, 0.0), line ( 0.0, 0.0, -1.7, 1.5), green(line ( -1.7, 1.5, 1.0, 2.3)), green(line ( 1.0, 2.3, 4.5, 2.3)), green(line ( 4.5, 2.3, 6.5, 1.0)), line ( 6.5, 1.0, 4.0, 0.0), line ( 0.0, 0.0, 2.25, 7.15), line ( -1.7, 1.5, 2.25, 7.15), green(line ( 1.0, 2.3, 2.25, 7.15)), green(line ( 4.5, 2.3, 2.25, 7.15)), line ( 6.5, 1.0, 2.25, 7.15), line ( 4.0, 0.0, 2.25, 7.15) )}}} <B>Figure 1e</B>. Hexagonal pyramid </TD> </TR> </TABLE> <H3>Formula for calculating the volume of pyramids</H3> <B>The volume of a pyramid</B> is {{{V}}} = {{{1/3}}}{{{S[base]}}}.{{{h}}}, where {{{S[base]}}} is the area of the base of the pyramid and {{{h}}} are the pyramid's height. <H3>My lessons on volume of pyramids in this site</H3> - <A HREF=http://www.algebra.com/algebra/homework/Volume/-Volume-of-pyramids.lesson>Volume of pyramids</A> under the topic <B>Volume, metric volume</B> of the section <B>Geometry</B>, and - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-volume-of-pyramids.lesson>Solved problems on volume of pyramids</A> under the topic <B>Geometry</B> of the section <B>Word problems</B>. <H3>Solved problems on volume of pyramids</H3><TABLE> <TR> <TD> Lesson <A HREF=http://www.algebra.com/algebra/homework/Volume/-Volume-of-pyramids.lesson>Volume of pyramids</A> - Find the volume of a regular pyramid with the square base if the height of the pyramid is of 12 cm and the measure of the base edge is of 10 cm. - Find the volume of a regular pyramid with the square base if the lateral edge of the pyramid has the same measure of 12 cm as the base edge has. Also find the angle between the lateral edge and the base plane of the pyramid. </TD> <TD> {{{drawing( 110, 110, -2.5, 3.5, -0.5, 3.8, line ( 0.0, 0.0, 3.0, 0.0), line ( 0.0, 0.0, -2.0, 0.8), green(line ( -2.0, 0.8, 1.0, 0.8)), green(line ( 1.0, 0.8, 3.0, 0.0)), line ( 0.0, 0.0, 0.5, 3.4), line ( 3.0, 0.0, 0.5, 3.4), green(line ( 1.0, 0.8, 0.5, 3.4)), line ( -2.0, 0.8, 0.5, 3.4), red(line ( 0.5, 3.4, 0.5, 0.4)) )}}} </TD> <TD> {{{drawing( 105, 100, -2.5, 3.5, -0.5, 3.8, line ( 0.0, 0.0, 3.0, 0.0), line ( 0.0, 0.0, -2.0, 0.8), green(line ( -2.0, 0.8, 1.0, 0.8)), green(line ( 1.0, 0.8, 3.0, 0.0)), line ( 0.0, 0.0, 0.5, 2.9), line ( 3.0, 0.0, 0.5, 2.9), green(line ( 1.0, 0.8, 0.5, 2.9)), line ( -2.0, 0.8, 0.5, 2.9) )}}} </TD> </TR> </TABLE><TABLE> <TR> <TD> - Find the volume of a regular hexagonal pyramid if its base edge is of 4 cm and the height of the pyramid is of 6 cm. - Find the volume of a regular tetrahedron if all its edges are of 10 cm long. </TD> <TD> {{{drawing( 119, 116, -2.5, 7.0, -0.8, 8.3, line ( 0.0, 0.0, 4.0, 0.0), line ( 0.0, 0.0, -1.7, 1.5), green(line ( -1.7, 1.5, 1.0, 2.3)), green(line ( 1.0, 2.3, 4.5, 2.3)), green(line ( 4.5, 2.3, 6.5, 1.0)), line ( 6.5, 1.0, 4.0, 0.0), line ( 0.0, 0.0, 2.25, 7.15), line ( -1.7, 1.5, 2.25, 7.15), green(line ( 1.0, 2.3, 2.25, 7.15)), green(line ( 4.5, 2.3, 2.25, 7.15)), line ( 6.5, 1.0, 2.25, 7.15), line ( 4.0, 0.0, 2.25, 7.15), red(line( 2.25, 1.2, 2.25, 7.15)) )}}} </TD> <TD> {{{drawing( 119, 112, -2.5, 4.2, -0.8, 8.2, line ( 0.0, 0.0, 4.0, 0.0), line ( 0.0, 0.0, -2.0, 1.5), green(line ( -2.0, 1.5, 4.0, 0.0)), line ( 0.667, 7.0, 0.0, 0.0), line ( 0.667, 7.0, 4.0, 0.0), line ( 0.667, 7.0, -2.0, 1.5) )}}} </TD> </TR> </TABLE> <TABLE> <TR> <TD> - Find the volume of a composite solid body of a "diamond" shape which comprises of two regular tetrahedrons joined face to face, if all their edges are of 4 cm long. - Find the volume of a body obtained from the regular tetrahedron with the edge measures of 10 cm after cutting off the part of the tetrahedron by the plane parallel to one of its faces in a way that the cutting plane bisects the three edges of the original tetrahedron (truncated regular tetrahedron). </TD> <TD>{{{drawing( 80, 110, -2.5, 4.5, -6.5, 8.0, line ( 0.0, 0.0, 4.0, 0.0), line ( 0.0, 0.0, -2.0, 1.5), green(line ( -2.0, 1.5, 4.0, 0.0)), line ( 0.667, 7.0, 0.0, 0.0), line ( 0.667, 7.0, 4.0, 0.0), line ( 0.667, 7.0, -2.0, 1.5), line ( 0.667, -5.4, 0.0, 0.0), line ( 0.667, -5.4, 4.0, 0.0), line ( 0.667, -5.4, -2.0, 1.5) )}}} </TD> <TD> {{{drawing( 125, 112, -2.5, 4.5, -0.8, 8.2, line ( 0.0, 0.0, 4.0, 0.0), line ( 0.0, 0.0, -2.0, 1.5), green(line ( -2.0, 1.5, 4.0, 0.0)), line ( 0.667, 7.0, 0.0, 0.0), line ( 0.667, 7.0, 4.0, 0.0), line ( 0.667, 7.0, -2.0, 1.5), line ( 0.333, 3.5, 2.333, 3.5), line ( 0.333, 3.5, -0.666, 4.25), line ( -0.666, 4.25, 2.333, 3.5), green(line ( 0.333, 3.5, 0.667, 7.0)), green(line ( -0.666, 4.25, 0.667, 7.0)), green(line ( 2.333, 3.5, 0.667, 7.0)) )}}} </TD> </TR> </TABLE><TABLE> <TR> <TD> Lesson <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-volume-of-pyramids.lesson>Solved problems on volume of pyramids</A> - Find the volume of a triangular pyramid <B>ABCD</B> if its edges issued from the vertex <B>A</B> are of 8 cm, 6 cm and 6 cm long and each of these three edges is perpendicular to the two others. - Find the volume of a rectangular pyramid <B>ABCDE</B> if its base <B>ABCD</B> is a square with the side measure of 6 cm and the lateral edge <B>AE</B> is perpendicular to the base plane and has the measure of 8 cm. </TD> <TD>{{{drawing( 112, 112, -4.0, 5.0, -0.5, 8.5, line ( -3.0, 1.5, 4.5, 0.0), green(line ( -3.0, 1.5, 2.0, 2.0)), green(line ( 2.0, 2.0, 4.5, 0.0)), line ( -3.0, 1.5, 2.0, 8.0), line ( 4.5, 0.0, 2.0, 8.0), green(line ( 2.0, 2.0, 2.0, 8.0)), locate ( 2.1, 2.8, A), locate (-3.4, 1.7, B), locate ( 4.7, 0.4, C), locate ( 2.2, 8.6, D) )}}} </TD> <TD> {{{drawing( 112, 119, -4.0, 5.0, -1.0, 8.5, line ( -3.0, 1.5, -0.5, -0.5), green(line ( -3.0, 1.5, 2.0, 2.0)), green(line ( 2.0, 2.0, 4.5, 0.0)), line ( -0.5, -0.5, 4.5, 0.0), line ( -3.0, 1.5, 2.0, 8.0), line ( 4.5, 0.0, 2.0, 8.0), green(line ( 2.0, 2.0, 2.0, 8.0)), line ( -0.5, -0.5, 2.0, 8.0), locate ( 2.1, 2.8, A), locate (-3.5, 1.7, B), locate (-1.2, -0.2, C), locate ( 4.6, 0.2, D), locate ( 2.1, 8.7, E) )}}} </TD> </TR> </TABLE><TABLE> <TR> <TD> - Find the volume of a regular pyramid with the square base if the lateral edge of the pyramid has the same measure of 10 cm as the the base edge has. Also find the angle between the lateral edge and the base of the pyramid. - Find the volume of a regular hexagonal pyramid if the base edge has the measure of 4 cm and the lateral edge of the pyramid is of 8 cm long. Also find the angle between the lateral edge and the base plane of the pyramid. </TD> <TD> {{{drawing( 110, 100, -2.5, 3.5, -0.5, 3.8, line ( 0.0, 0.0, 3.0, 0.0), line ( 0.0, 0.0, -2.0, 0.8), green(line ( -2.0, 0.8, 1.0, 0.8)), green(line ( 1.0, 0.8, 3.0, 0.0)), line ( 0.0, 0.0, 0.5, 2.9), line ( 3.0, 0.0, 0.5, 2.9), green(line ( 1.0, 0.8, 0.5, 2.9)), line ( -2.0, 0.8, 0.5, 2.9), red(line ( 0.5, 2.9, 0.5, 0.4)), red(line ( 0.5, 0.4, 0.0, 0.0)), arc (0, 0, 1.0, 1.0, 280, 323) )}}} </TD> <TD> {{{drawing( 119, 116, -2.5, 7.0, -0.8, 8.3, line ( 0.0, 0.0, 4.0, 0.0), line ( 0.0, 0.0, -1.7, 1.5), green(line ( -1.7, 1.5, 1.0, 2.3)), green(line ( 1.0, 2.3, 4.5, 2.3)), green(line ( 4.5, 2.3, 6.5, 1.0)), line ( 6.5, 1.0, 4.0, 0.0), line ( 0.0, 0.0, 2.25, 7.5), line ( -1.7, 1.5, 2.25, 7.5), green(line ( 1.0, 2.3, 2.25, 7.5)), green(line ( 4.5, 2.3, 2.25, 7.5)), line ( 6.5, 1.0, 2.25, 7.5), line ( 4.0, 0.0, 2.25, 7.5), red(line( 2.25, 1.2, 2.25, 7.5)), red(line( 2.25, 1.2, 0.00, 0.0)), arc( 0.00, 0.00, 1.6, 1.6, 283, 330) )}}} </TD> </TR> </TABLE><TABLE> <TR> <TD> - Find the volume of a composite solid body of a "diamond" shape which comprises of two regular rectangular pyramids with square bases joined base to base, if all their edges are of 4 cm. - Find the volume of a body obtained from a regular rectangular pyramid with the edge measures of 10 cm for all edges after cutting off the part of the pyramid by the plane parallel to the base in a way that the cutting plane bisects the four lateral edges of the original pyramid (truncated regular rectangular pyramid). </TD> <TD>{{{drawing( 90, 110, -2.5, 3.5, -2.5, 3.3, line ( 0.0, 0.0, 3.0, 0.0), line ( 0.0, 0.0, -2.0, 0.8), green(line ( -2.0, 0.8, 1.0, 0.8)), green(line ( 1.0, 0.8, 3.0, 0.0)), line ( 0.0, 0.0, 0.5, 2.9), line ( 3.0, 0.0, 0.5, 2.9), green(line ( 1.0, 0.8, 0.5, 2.9)), line ( -2.0, 0.8, 0.5, 2.9), line ( 0.0, 0.0, 0.5, -2.1), line ( 3.0, 0.0, 0.5, -2.1), green(line ( 1.0, 0.8, 0.5, -2.1)), line ( -2.0, 0.8, 0.5, -2.11) )}}} </TD> <TD> {{{drawing( 110, 100, -2.5, 3.5, -0.5, 3.8, line ( 0.0, 0.0, 3.0, 0.0), line ( 0.0, 0.0, -2.0, 0.8), green(line ( -2.0, 0.8, 1.0, 0.8)), green(line ( 1.0, 0.8, 3.0, 0.0)), line ( 0.0, 0.0, 0.25, 1.45), line ( 3.0, 0.0, 1.75, 1.45), green(line ( 1.0, 0.8, 0.5, 2.9)), line ( -2.0, 0.8, -0.75, 1.85), line ( 0.25, 1.45, 1.75, 1.45), line ( 0.25, 1.45, -0.75, 1.85), line ( -0.75, 1.85, 0.75, 1.85), line ( 0.75, 1.85, 1.75, 1.45), green(line ( 0.25, 1.45, 0.5, 2.9)), green(line ( 1.75, 1.45, 0.5, 2.9)), green(line ( 0.75, 1.85, 0.5, 2.9)), green(line ( -0.75, 1.85, 0.5, 2.9)) )}}} </TD> </TR> </TABLE> My lessons on volume of pyramids and other 3D solid bodies in this site are <TABLE> <TR> <TD> <B>Lessons on volume of prisms</B> <A HREF=http://www.algebra.com/algebra/homework/Volume/_Volume-of-prisms.lesson>Volume of prisms</A> <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-volume-of-prisms.lesson>Solved problems on volume of prisms</A> <A HREF=http://www.algebra.com/algebra/homework/Volume/OVERVIEW-of-LESSONS-on-volume-of-prisms.lesson>Overview of lessons on volume of prisms</A> </TD> <TD> <B>Lessons on volume of pyramids</B> <A HREF=http://www.algebra.com/algebra/homework/Volume/_Volume-of-pyramids.lesson>Volume of pyramids</A> <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-volume-of-pyramids.lesson>Solved problems on volume of pyramids</A> Overview of lessons on volume of pyramids </TD> </TR> </Table><TABLE> <TR> <TD> <B>Lessons on volume of cylinders</B> <A HREF=http://www.algebra.com/algebra/homework/Volume/_Volume-of-cylinders.lesson>Volume of cylinders</A> <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-volume-of-cylinders.lesson>Solved problems on volume of cylinders</A> <A HREF=http://www.algebra.com/algebra/homework/Volume/OVERVIEW-of-LESSONS-on-Volume-of-cylinders.lesson>Overview of lessons on volume of cylinders</A> </TD> <TD> <B>Lessons on volume of cones</B> <A HREF=http://www.algebra.com/algebra/homework/Volume/Volume-of-cones.lesson>Volume of cones</A> <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-Volume-of-cones.lesson>Solved problems on volume of cones</A> <A HREF=http://www.algebra.com/algebra/homework/Volume/OVERVIEW-of-LESSONS-on-Volume-of-cones.lesson>Overview of lessons on volume of cones</A> </TD> <TD> <B>Lessons on volume of spheres</B> <A HREF=http://www.algebra.com/algebra/homework/Volume/Volume-of-spheres.lesson>Volume of spheres</A> <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-Volume-of-spheres.lesson>Solved problems on volume of spheres</A> <A HREF=http://www.algebra.com/algebra/homework/Volume/OVERVIEW-of-LESSONS-on-Volume-of-spheres.lesson>Overview of lessons on volume of spheres</A> </TD> </TR> </Table> To navigate over all topics/lessons of the Online Geometry Textbook use this file/link <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.
