Lesson OVERVIEW of LESSONS on Surface Area of PYRAMIDS
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<H2>OVERVIEW of LESSONS on Surface Area of Pyramids</H2>For your convenience, this file contains - the list of my lessons on surface area of pyramids in this site, - the major formulas for calculating the surface area 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>Major formulas for calculating the surface area of pyramids</H3> 1. <B>The lateral surface area of a pyramid</B> is {{{S[lateral]}}} = {{{1/2}}}( {{{a[1]*l[1]}}} + {{{a[2]*l[2]}}} + . . . + {{{a[n]*l[n]}}} ), where {{{a[i]}}} are the base edge measures and {{{l[i]}}} are the slant heights, i = 1, 2, . . . , n. 2. <B>The lateral surface area</B> <B>of a <U>regular pyramid</U></B> is {{{S[lateral]}}} = {{{1/2}}}{{{n*a*l}}} = {{{1/2}}}{{{P*l}}}, where {{{a}}} is the base edge measure, {{{l}}} is the slant height and {{{P}}} is the perimeter of the base polygon. 3. <B>The total surface area</B> <B>S</B> <B>of a pyramid</B> is {{{S}}} = {{{S[lateral] + S[base]}}}, where {{{S[lateral]}}} is the lateral surface area of the pyramid and {{{S[base]}}} is the pyramid base area. <H3>My lessons on surface area of pyramids in this site</H3> - <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Surface-area-of-pyramids.lesson>Surface area of pyramids</A> under the topic <B>Area and surface area</B> of the section <B>Geometry</B>, and - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-surface-area-of-pyramids.lesson>Solved problems on surface area of pyramids</A> under the topic <B>Geometry</B> of the section <B>Word problems</B>. <H3>Solved problems on surface area of pyramids</H3>Lesson <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Surface-area-of-pyramids.lesson>Surface area of pyramids</A> - Find the lateral surface area 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 surface area of a regular tetrahedron if all its edges are of 10 cm long. - Find the lateral surface area of a regular hexagonal pyramid if its base edge is of 4 cm and the height of the pyramid is of 6 cm.<TABLE> <TR> <TD> - Find the surface area of a combined 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 lateral surface area 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. </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-surface-area-of-pyramids.lesson>Solved problems on surface area of pyramids</A> - Find the surface area 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 surface area 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 lateral surface area 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. Then find the total surface area of the pyramid. Also find the angle between the lateral edge and the base of the pyramid. - Find the lateral surface area of a regular hexagonal pyramid if the base edge has the measure of 4 cm and the height of the pyramid is of 2 cm. Then find the total surface area of the pyramid. Also find the angle between the lateral face plane 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, 105, -2.5, 7.0, -0.8, 7.4, 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)), 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, 3.2), line ( -1.7, 1.5, 2.25, 3.2), green(line ( 1.0, 2.3, 2.25, 3.2)), line ( 4.5, 2.3, 2.25, 3.2), line ( 6.5, 1.0, 2.25, 3.2), line ( 4.0, 0.0, 2.25, 3.2), red(line( 2.25, 1.2, 2.25, 3.2)), red(line( 2.25, 1.2, -0.85, 0.75)), red(line( -0.85, 0.75, 2.25, 3.2)), arc(-0.85, 0.75, 2.4, 2.4, 317, 350) )}}} </TD> </TR> </TABLE> <TABLE> <TR> <TD> - Find the surface area of a combined 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 lateral surface area 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. </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 surface area of pyramids and other 3D solid bodies in this site are <TABLE> <TR> <TD> <B>Lessons on surface area of prisms</B> <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Surface-area-of-prisms.lesson>Surface area of prisms</A> <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-surface-area-of-prisms.lesson>Solved problems on surface area of prisms</A> <A HREF=http://www.algebra.com/algebra/homework/Surface-area/OVERVIEW-of-LESSONS-on-surface-area-of-prisms.lesson>Overview of lessons on surface area of prisms</A> </TD> <TD> <B>Lessons on surface area of pyramids</B> <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Surface-area-of-pyramids.lesson>Surface area of pyramids</A> <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-surface-area-of-pyramids.lesson>Solved problems on surface area of pyramids</A> Overview of lessons on surface area of pyramids </TD> </TR> </Table><TABLE> <TR> <TD> <B>Lessons on surface area of cylinders</B> <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Surface-area-of-cylinders.lesson>Surface area of cylinders</A> <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-surface-area-of-cylinders.lesson>Solved problems on surface area of cylinders</A> <A HREF=http://www.algebra.com/algebra/homework/Surface-area/OVERVIEW-of-LESSONS-on-surface-area-of-cylinders.lesson>Overview of lessons on surface area of cylinders</A> </TD> <TD> <B>Lessons on surface area of cones</B> <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Surface-area-of-cones.lesson>Surface area of cones</A> <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-surface-area-of-cones.lesson>Solved problems on surface area of cones</A> <A HREF=http://www.algebra.com/algebra/homework/Surface-area/OVERVIEW-of-LESSONS-on-surface-area-of-cones.lesson>Overview of lessons on surface area of cones</A> </TD> <TD> <B>Lessons on surface area of spheres</B> <A HREF=http://www.algebra.com/algebra/homework/Surface-area/Surface-area-of-spheres.lesson>Surface area of spheres</A> <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-surface-area-of-spheres.lesson>Solved problems on surface area of spheres</A> <A HREF=http://www.algebra.com/algebra/homework/Surface-area/OVERVIEW-of-LESSONS-on-surface-area-of-spheres.lesson>Overview of lessons on surface area 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>.
