Lesson Volume of prisms
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<H2>Volume of prisms</H2> Prisms are solid bodies with flat faces that have two congruent parallel faces and a set of parallel edges that connect corresponding vertices of the two parallel faces. Figures <B>1a</B> - <B>1e</B> present the examples of prisms. <TABLE> <TR> <TD> {{{drawing( 200, 150, -4.0, 4.0, -0.5, 5.5, line ( 0.0, 0.0, 3.0, 0.0), line ( 0.0, 0.0, 0.0, 3.0), line ( 0.0, 0.0, -2.0, 0.8), line ( 0.0, 3.0, 3.0, 3.0), line ( 3.0, 3.0, 3.0, 0.0), line ( 0.0, 3.0, -2.0, 3.8), line ( -2.0, 3.8, 1.0, 3.8), line ( -2.0, 3.8, -2.0, 0.8), green(line ( 1.0, 3.8, 1.0, 0.8)), line ( 1.0, 3.8, 3.0, 3.0), green(line ( -2.0, 0.8, 1.0, 0.8)), green(line ( 1.0, 0.8, 3.0, 0.0)), locate( 1.2, 0.1, a), locate(-1.5, 0.6, a), locate(-0.4, 2.1, a) )}}} <B>Figure 1a</B>. A cube </TD> <TD> {{{drawing( 250, 150, -5.0, 5.0, -0.5, 5.5, line ( 0.0, 0.0, 4.0, 0.0), line ( 0.0, 0.0, 0.0, 3.0), line ( 0.0, 0.0, -4.0, 1.5), line ( 0.0, 3.0, 4.0, 3.0), line ( 4.0, 3.0, 4.0, 0.0), line ( 0.0, 3.0, -4.0, 4.5), line ( -4.0, 4.5, -4.0, 1.5), line ( -4.0, 4.5, -0.5, 4.5), line ( -4.0, 4.5, -4.0, 1.5), green(line ( -0.5, 4.5, -0.5, 1.5)), line ( -0.5, 4.5, 4.0, 3.0), green(line ( -4.0, 1.5, -0.5, 1.5)), green(line ( -0.5, 1.5, 4.0, 0.0)), locate( 1.7, 0.05, u), locate(-2.3, 0.80, v), locate( 0.15, 2.20, w) )}}} <B>Figure 1b</B>. A rectangular prism </TD> <TD> {{{drawing( 225, 225, -4.0, 5.0, -0.5, 8.5, line ( 0.0, 0.0, 4.0, 0.0), line ( 0.0, 0.0, 0.0, 7.0), line ( 0.0, 0.0, -3.0, 1.5), line ( 4.0, 7.0, 4.0, 0.0), line ( 4.0, 7.0, 0.0, 7.0), line ( -3.0, 8.5, -3.0, 1.5), line ( -3.0, 8.5, 0.0, 7.0), line ( -3.0, 8.5, 4.0, 7.0), green(line ( -3.0, 1.5, 4.0, 0.0)) )}}} <B>Figure 1c</B>. A triangular prism </TD> <TD> {{{drawing( 225, 225, -4.0, 5.0, -0.5, 8.5, line ( 0.0, 0.0, 4.0, 0.0), line ( 0.0, 0.0, -3.0, 1.5), green(line ( -3.0, 1.5, 2.0, 2.0)), green(line ( 2.0, 2.0, 4.0, 0.0)), line ( 0.0, 6.0, 4.0, 6.0), line ( 0.0, 6.0, -3.0, 7.5), line ( -3.0, 7.5, 2.0, 8.0), line ( 2.0, 8.0, 4.0, 6.0), line ( 0.0, 0.0, 0.0, 6.0), line ( -3.0, 1.5, -3.0, 7.5), green(line ( 2.0, 2.0, 2.0, 8.0)), line ( 4.0, 0.0, 4.0, 6.0) )}}} <B>Figure 1d</B>. A quadrilatelar prism </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, 6.0, 4.0, 6.0), line ( 0.0, 6.0, -1.7, 7.5), line ( -1.7, 7.5, 1.0, 8.3), line ( 1.0, 8.3, 4.5, 8.3), line ( 4.5, 8.3, 6.5, 7.0), line ( 6.5, 7.0, 4.0, 6.0), line ( 0.0, 0.0, 0.0, 6.0), line ( -1.7, 1.5, -1.7, 7.5), green(line ( 1.0, 2.3, 1.0, 8.3)), green(line ( 4.5, 2.3, 4.5, 8.3)), line ( 6.5, 1.0, 6.5, 7.0), line ( 4.0, 0.0, 4.0, 6.0) )}}} <B>Figure 1e</B>. A hexagonal prism </TD> </TR> </TABLE> In the school geometry, only <B>upright prisms</B> are considered. They are prisms that have each lateral face plane perpendicular to the plane (to the planes) of the bases. It means, in particular, that each lateral face of an upright prism is a rectangle. It also means that each lateral edge of the upright prism, i.e. an intersection of two adjacent lateral faces, is perpendicular to the planes of the bases. All lateral edges of an upright prism are congruent. Any of these edges is (is called) the <B>height of a prism</B>. The bases of a prism are parts of their planes restricted by polygons. Depending on the shape of that polygons, the prisms can be called <B>triangular prisms</B>, or <B>rectangular prisms</B>, or <B>pentagonal</B>, <B>hexagonal</B> and so on. This lesson is focused on calculating the volume of prisms. <H3>Major formulas for calculating the volume of prisms</H3><TABLE> <TR> <TD> 1. <B>Volume</B> <B>V</B> <B>of a <U>cube</U></B> with the edge measure <B>a</B> is {{{V}}} = {{{a^3}}}. 2. <B>Volume</B> <B>V</B> <B>of a <U>rectangular prism</U></B> with the edge measures <B>u</B>, <B>v</B> and <B>w</B> is {{{V}}} = {{{uvw)}}}. 3. <B>Volume</B> <B>V</B> <B>of an <U>upright prism</U></B> is {{{V}}} = {{{S*h}}}, where <B>S</B> is the area of the prism base and <B>h</B> is the prism height. </TD> <TD> (1) (2) (3) </TD> </TR> </TABLE> All the formulas (1), (2) and (3) are the direct consequences of the base postulates for volume of the lesson <A HREF=http://www.algebra.com/algebra/homework/Volume/WHAT-IS-volume.lesson>What is volume?</A> under the topic <B>Volume, Metric volume</B> of the section <B>Geometry</B> in this site. <H3>Example 1</H3><TABLE> <TR> <TD>Find the volume of a cube with the edge measure of 5 cm. <B>Solution</B> The volume of a cube with the side measure {{{a}}} is {{{a^3}}} cubic units. So, in our case the volume of the cube is {{{V}}} = {{{5^3}}} = {{{125}}} {{{cm^3}}}. <B>Answer</B>. The volume of the cube is {{{V}}} = {{{125}}} {{{cm^3}}}. </TD> <TD> {{{drawing( 200, 150, -4.0, 4.0, -0.5, 5.5, line ( 0.0, 0.0, 3.0, 0.0), line ( 0.0, 0.0, 0.0, 3.0), line ( 0.0, 0.0, -2.0, 0.8), line ( 0.0, 3.0, 3.0, 3.0), line ( 3.0, 3.0, 3.0, 0.0), line ( 0.0, 3.0, -2.0, 3.8), line ( -2.0, 3.8, 1.0, 3.8), line ( -2.0, 3.8, -2.0, 0.8), green(line ( 1.0, 3.8, 1.0, 0.8)), line ( 1.0, 3.8, 3.0, 3.0), green(line ( -2.0, 0.8, 1.0, 0.8)), green(line ( 1.0, 0.8, 3.0, 0.0)), locate( 1.2, 0.05, 5), locate(-1.5, 0.55, 5), locate(-0.4, 2.10, 5) )}}} <B>Figure 2</B>. To the <B>Example 1</B> </TD> </TR> </TABLE> <H3>Example 2</H3>Find the volume of a rectangular prism which has the edge measures of 8 cm, 10 cm and 6 cm. <TABLE> <TR> <TD> <B>Solution</B> The volume of a rectangular prism with the side measures {{{u}}}, {{{v}}} and {{{w}}} is {{{u*v*w}}} cubic units. So, in our case the volume of the rectangular prism is {{{V}}} = {{{8*10*6}}} = {{{480}}} {{{cm^3}}}. <B>Answer</B>. The volume of the rectangular prism is {{{V}}} = {{{480}}} {{{cm^3}}}. </TD> <TD> {{{drawing( 250, 150, -5.0, 5.0, -0.5, 5.5, line ( 0.0, 0.0, 4.0, 0.0), line ( 0.0, 0.0, 0.0, 3.0), line ( 0.0, 0.0, -4.0, 1.5), line ( 0.0, 3.0, 4.0, 3.0), line ( 4.0, 3.0, 4.0, 0.0), line ( 0.0, 3.0, -4.0, 4.5), line ( -4.0, 4.5, -4.0, 1.5), line ( -4.0, 4.5, -0.5, 4.5), line ( -4.0, 4.5, -4.0, 1.5), green(line ( -0.5, 4.5, -0.5, 1.5)), line ( -0.5, 4.5, 4.0, 3.0), green(line ( -4.0, 1.5, -0.5, 1.5)), green(line ( -0.5, 1.5, 4.0, 0.0)), locate( 1.0, 0.6, u=8), locate(-3.5, 0.8, v=10), locate( 0.15, 2.3, w=6) )}}} <B>Figure 3</B>. To the <B>Example 2</B> </TD> </TR> </TABLE> <H3>Example 3</H3>Find the volume of a triangular prism if its base is an equilateral triangle with the side measure of 4 cm and the height of the prism is of 10 cm. <TABLE> <TR> <TD> <B>Solution</B> First, the area of the triangle at the base is {{{S[base]}}} = {{{1/2}}}.{{{4}}}.{{{4}}}{{{sqrt(3)/2}}} = {{{4*sqrt(3)}}} = {{{4*1.732}}} = 6.928 {{{cm^2}}} (approximately). Now, the volume of the prism is {{{V}}} = {{{S[base]*h}}} = 6.928*10 = = 69.28 {{{cm^3}}} (approximately). <B>Answer</B>. The volume of the prism is 69.28 {{{cm^3}}} (approximately). </TD> <TD> {{{drawing( 250, 225, -5.0, 5.0, -0.5, 8.5, line ( 0.0, 0.0, 4.0, 0.0), line ( 0.0, 0.0, 0.0, 7.0), line ( 0.0, 0.0, -2.0, 1.5), line ( 4.0, 7.0, 4.0, 0.0), line ( 4.0, 7.0, 0.0, 7.0), line ( -2.0, 8.5, -2.0, 1.5), line ( -2.0, 8.5, 0.0, 7.0), line ( -2.0, 8.5, 4.0, 7.0), green(line ( -2.0, 1.5, 4.0, 0.0)), locate( 0.7, -0.1, a=4), locate(-2.4, 1.0, b=4), locate( 0.15, 4.0, w=10), locate( 1.0, 1.3, c=4) )}}} <B>Figure 4</B>. To the <B>Example 3</B> </TD> </TR> </TABLE> <H3>Example 4</H3>Find the volume of a hexagonal prism if its base is a regular hexagon with the side measure of 4 cm and the height of the prism is of 10 cm. <TABLE> <TR> <TD> <B>Solution</B> First, the area of the regular hexagon at the base is {{{S[base]}}} = {{{6}}}.({{{1/2}}}.{{{4}}}.{{{4}}}{{{sqrt(3)/2}}}) = {{{24sqrt(3)}}} = {{{24*1.732}}} = {{{41.569}}} {{{cm^2}}} (approximately). Now, the volume of the prism is {{{V}}} = {{{S[base]*h}}} = 41.569*10 = = 415.69 {{{cm^3}}} (approximately). <B>Answer</B>. The volume of the prism is 415.69 {{{cm^2}}} (approximately). </TD> <TD> {{{drawing( 250, 238, -2.5, 7.5, -0.5, 9.0, 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, 6.0, 4.0, 6.0), line ( 0.0, 6.0, -1.7, 7.5), line ( -1.7, 7.5, 1.0, 8.3), line ( 1.0, 8.3, 4.5, 8.3), line ( 4.5, 8.3, 6.5, 7.0), line ( 6.5, 7.0, 4.0, 6.0), line ( 0.0, 0.0, 0.0, 6.0), line ( -1.7, 1.5, -1.7, 7.5), green(line ( 1.0, 2.3, 1.0, 8.3)), green(line ( 4.5, 2.3, 4.5, 8.3)), line ( 6.5, 1.0, 6.5, 7.0), line ( 4.0, 0.0, 4.0, 6.0), locate( 1.8, 0.0, 4), locate(-1.5, 1.0, 4), locate( 5.3, 0.6, 4), locate( 0.05, 4.0, 10) )}}} <B>Figure 5</B>. To the <B>Example 4</B> </TD> </TR> </TABLE> My lessons on volume of prisms and other 3D solid bodies in this site are <TABLE> <TR> <TD> <B>Lessons on volume of prisms</B> Volume of prisms <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> <A HREF=http://www.algebra.com/algebra/homework/Volume/OVERVIEW-of-LESSONS-on-volume-of-pyramids.lesson>Overview of lessons on volume of pyramids</A> </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>.
