Lesson Volume of cylinders
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<H2>Volume of cylinders</H2> <B>Cylinder</B> is a <B>3D</B> solid body which is bounded by two congruent circular bases in parallel planes and the lateral surface consisting of parallel straight segments that connect the corresponding points at the base circles. The <B>Figure 1a</B> represents an example of a cylinder. You may think that the circle <B>O2</B> is obtained from the circle <B>O1</B> by the parallel transfer along the straight line <B>O1O2</B> connecting the centers of the circles. Then the corresponding points at the base circles are those obtained by this transfer. <TABLE> <TR> <TD> {{{drawing( 220, 225, -5.5, 5.5, -6.0, 5.4, ellipse( 1.0, -3.0, 8.0, 4.0), ellipse(-1.0, 2.0, 8.0, 4.0), line( -3.0, -3.0, -5.0, 2.0), line( 5.0, -3.0, 3.0, 2.0), locate ( -5.4, -5.1, base), line ( -4.6, -5.0, -1.0, -3.5), locate ( 2.7, 5.3, lateral), locate ( 2.7, 4.7, surface), line ( 3.9, 3.9, 3.3, 0.5), locate ( -5.4, 5.0, base), line ( -4.6, 4.1, -1.5, 2.5), green(line ( 1.0, -3, -1.0, 2)), circle ( 1.0, -3, 0.1, 0.1), locate ( 1.0, -2.4, O1), circle (-1.0, 2, 0.1, 0.1), locate (-1.0, 2.6, O2) )}}} <B>Figure 1a</B>. Cylinder (general definition) </TD> <TD> {{{drawing( 220, 225, -5.5, 5.5, -6.0, 5.4, ellipse( 0.0, -3.0, 8.0, 4.0), ellipse( 0.0, 2.0, 8.0, 4.0), line( -3.92, -3.0, -3.92, 2.0), line( 4.0, -3.0, 4.0, 2.0), locate ( -5.4, -5.1, base), line ( -4.6, -5.0, -1.0, -3.5), locate ( 2.7, 5.3, lateral), locate ( 2.7, 4.7, surface), line ( 3.9, 3.9, 3.3, 0.5), locate ( -5.4, 5.0, base), line ( -4.6, 4.1, -1.0, 2.5), green(line ( 0.0, -3.5, 0.0, 2.5)), circle ( 0.0, -3, 0.1, 0.1), locate ( 0.15, -3.0, O1), circle ( 0.0, 2, 0.1, 0.1), locate ( 0.15, 2.6, O2), locate ( 4.15, 0.2, h), locate (-4.45, 0.2, h), locate ( 3.3, -5.1, height), green(line ( 4.35, -4.9, 4.35, -0.6)) )}}} <B>Figure 1b</B>. Right circular cylinder </TD> </TR> </TABLE> In the school geometry, only the <B>right circular cylinders</B> are considered. They are cylinders that have the axis (the straight line connecting the centers of the circles) perpendicular to the plane (to the planes) of the bases (<B>Figure 1b</B>). It means, as a consequence, that each lateral generatrix of a right circular cylinder is perpendicular to the planes of the bases. All lateral generatrices of an right circular cylinder are congruent. Any of these generatrices is (is called) the <B>height of a cylinder</B>. You may think a right circular cylinder as a <B>3D</B> solid body which is swept by a rectangle rotating in <B>3D</B> space around some of its sides as an sxis when the rectangle makes the full revolution (the full turn). This lesson is focused on calculating the volume of the right circular cylinders. <H3>Formula for calculating the volume of cylinders</H3> <B>The volume of a cylinder</B> is {{{V}}} = {{{pi}}}{{{r^2}}}{{{h}}} = {{{S[base]}}}{{{h}}}, where {{{r}}} is the radius of the cylinder, {{{h}}} is its height, and {{{S[base]}}}= {{{pi}}}{{{r^2}}} is the base area (the area of the circle at the base). <H3>Example 1</H3>Find the volume of a cylinder if its radius is of 5 cm and the height is of 4 cm. <B>Solution</B> The volume of the cylinder is {{{V}}} = {{{pi}}}{{{r^2}}}{{{h}}} = {{{3.14159}}}*{{{5^2}}}*{{{4}}} = 3.14159*25*4 = 3.14159*100 = 314.159 {{{cm^2}}} (approximately). <B>Answer</B>. The volume of the cylinder is 314.159 {{{cm^2}}} (approximately). <H3>Example 2</H3>Two cylinders are joined in a way that the base of one cylinder is overposed on the base of the other as shown in the <B>Figure 2a</B>. The radius of one cylinder is 5 cm and the height is 2 cm. The radius of the other cylinder is 2 cm and the height is 5 cm. Find the volume of the composite body. <TABLE> <TR> <TD> <B>Solution</B> The <B>Figure 2b</B> represents the side view of the two cylinders. The common axis is shown in blue in the Figures <B>2a</B> and <B>2b</B>. The volume under consideration is composed of the volume of the first cylinder and the volume of the second cylinder: {{{V}}} = {{{pi*r[1]^2}}}{{{h[1]}}} + {{{pi*r[2]^2*h[2]}}} = {{{pi}}}.{{{5^2}}}.{{{2}}} + {{{pi*2^2}}}.{{{5}}} = {{{pi}}}.( {{{25*2 + 4*5}}} ) = = {{{70}}}{{{pi}}} = 70*3.14159 = 219.91 {{{cm^2}}}. <B>Answer</B>. The volume of the composite body is 219.91 {{{cm^2}}} (approximately). </TD> <TD> {{{drawing( 220, 228, -5.5, 5.5, -6.0, 5.4, ellipse( 0.0, -3.0, 10.0, 6.0), ellipse( 0.0, -1.0, 10.0, 6.0), line( -5.0, -3.0, -5.0, -1.0), line( 5.0, -3.0, 5.0, -1.0), ellipse( 0.0, -1.0, 4.0, 2.4), ellipse( 0.0, 4.0, 4.0, 2.4), line( 2.0, -1.0, 2.0, 4.0), line(-2.0, -1.0, -2.0, 4.0), blue(line( 0.0, 4.0, 0.0, -3.0)), locate (-3.0, -1.5, 1), locate (-1.1, 2.9, 2) )}}} <B>Figure 2a</B>. To the <B>Example 2</B> </TD> <TD> {{{drawing( 220, 228, -5.5, 5.5, -6.0, 5.4, line( -5.0, -3.0, 5.0, -3.0), line (-5.0, -1.0, 5.0, -1.0), line (-5.0, -3.0, -5.0, -1.0), line ( 5.0, -3.0, 5.0, -1.0), blue(line ( 0.0, -3.5, 0.0, 4.5)), line(-2.0, -1.0, -2.0, 4.0), line( 2.0, -1.0, 2.0, 4.0), line (-2, 4, 2, 4), locate (-3.0, -1.5, 1), locate (-1.1, 2.7, 2) )}}} <B>Figure 2b</B>. Side view of the two cylinders </TD> </TR> </TABLE> <B>Note</B>. The assumption that the cylinders are co-axial is not necessary. The result is valid for non-axial cylinders too. <H3>Example 3</H3>Find the volume of the solid body concluded between two co-axial cylindrical surfaces (<B>Figure 3</B>) of the radii of 10 cm and 5 cm respectively if the common height of the two cylindrical shells is of 8 cm.<TABLE> <TR> <TD> <B>Solution</B> The <B>Figure 3</B> represents the solid body concluded between two co-axial cylindrical surfaces. Their common axis is shown in blue in this <B>Figure</B>. The volume under consideration is the volume of the larger cylinder minus the volume of the smaller one, i.e. {{{V}}} = {{{pi*10^2*8}}} - {{{pi*5^2*8}}} = {{{pi*(100 - 25)*8}}} = 3.14159*75*8 = 1884.95 {{{cm^3}}} (approximately). <B>Answer</B>. The volume of the solid body under consideration is 1884.95 {{{cm^3}}} (approximately). </TD> <TD> {{{drawing( 220, 228, -5.5, 5.5, -6.0, 5.4, green(circle( 0, 0, 0.1, 0.1)), ellipse( 0.0, -3.0, 10.0, 4.0), ellipse( 0.0, 3.0, 10.0, 4.0), ellipse( 0.0, 3.0, 4.0, 1.6), green(ellipse( 0.0, -3.0, 4.0, 1.6)), line( 5.0, 3.0, 5.0, -3.0), line(-5.0, 3.0, -5.0, -3.0), green(line( 2.0, 3.0, 2.0, -3.0)), green(line(-2.0, 3.0, -2.0, -3.0)), blue(line( 0.0, 3.0, 0.0, -3.0)) )}}} <B>Figure 3</B>. To the <B>Example 3</B> </TD> </TR> </TABLE> <H3>Example 4</H3>Four through cylindrical holes are made in the solid cylinder parallel to its axis of symmetry (<B>Figure 4</B>). Find the volume of the obtained solid body if the diameter of the original cylinder is 10 cm, its height is 8 cm and the diameter of each hole is 2 cm. <TABLE> <TR> <TD> <B>Solution</B> The volume of the original solid cylinder is {{{V}}} = {{{pi}}}{{{r^2}}}{{{h}}} = {{{pi}}}*{{{5^2}}}*{{{8}}} = {{{200*pi}}}. The volume of each of four holes is {{{V[hole]}}} = {{{pi}}}*{{{1^2}}}*{{{8}}} = {{{8*pi}}}. Hence, the volume of the solid body under consideration is {{{V}}} - {{{4*V[hole]}}} = {{{200*pi}}} - {{{4*8*pi}}} = {{{168*pi}}} = 527.52 {{{cm^3}}} (approximately). <B>Answer</B>. The volume of the solid body under consideration is 527.52 {{{cm^3}}} (approximately). </TD> <TD> {{{drawing( 220, 228, -5.5, 5.5, -6.0, 5.4, green(circle( 0, 0, 0.1, 0.1)), ellipse( 0.0, 3.0, 10.0, 4.0), ellipse( 0.0, -3.0, 10.0, 4.0), line( 5.0, 3.0, 5.0, -3.0), line(-5.0, 3.0, -5.0, -3.0), blue(line( 0.0, 3.0, 0.0, -3.0)) ellipse( 2.5, 2.2, 2.0, 0.8), green(ellipse( 2.5, -3.8, 2.0, 0.8)), green(line( 1.5, 2.2, 1.5, -3.8)), green(line( 3.5, 2.2, 3.5, -3.8)), blue(line( 2.5, 2.2, 2.5, -3.8)), ellipse( 2.5, 3.8, 2.0, 0.8), green(line( 1.5, 3.8, 1.5, 2.6)), green(line( 3.5, 3.8, 3.5, 2.6)), blue(line( 2.5, 3.8, 2.5, 3.0)) ellipse(-2.5, 2.2, 2.0, 0.8), green(ellipse(-2.5, -3.8, 2.0, 0.8)), green(line(-1.5, 2.2, -1.5, -3.8)), green(line(-3.5, 2.2, -3.5, -3.8)), blue(line(-2.5, 2.2, -2.5, -3.8)), ellipse(-2.5, 3.8, 2.0, 0.8), green(line(-1.5, 3.8, -1.5, 2.6)), green(line(-3.5, 3.8, -3.5, 2.6)), blue(line(-2.5, 3.8, -2.5, 3.0)) )}}} <B>Figure 4</B>. To the <B>Example 4</B> </TD> </TR> </TABLE> <H3>Example 5</H3>A through cylindrical hole is made in a rectangular prism (rectangular box) of dimensions 6x8x10 cm along its axis of symmetry parallel to the shortest edge (<B>Figure 5</B>). Find the volume of the obtained solid body if the diameter of the hole is 2 cm. <TABLE> <TR> <TD> <B>Solution</B> The volume of the original rectangular prism is {{{V[prism]}}} = {{{6}}}.{{{8}}}.{{{10}}} = 480{{{cm^3}}}. The volume of the cylindrical hole is {{{V[hole]}}} = {{{pi}}}{{{r^2}}}{{{h}}} = {{{pi}}}*{{{1^2}}}*{{{6}}} = {{{6*pi}}}. Hence, the volume of the solid body under consideration is {{{V}}} = {{{V[prism]}}} - {{{V[hole]}}} = {{{480}}} - {{{6*pi}}} = {{{480}}} - {{{6*3.14}}} = 461.16 {{{cm^3}}} (approximately). <B>Answer</B>. The volume of the solid body under consideration is 461.16 {{{cm^3}}} (approximately). </TD> <TD> {{{drawing( 300, 180, -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), line ( -0.5, 4.5, 4.0, 3.0), locate( 1.0, 0.6, u=8), locate(-3.5, 0.8, v=10), locate( 0.15, 2.0, w=6), ellipse(-0.2, 3.7, 1.0, 0.3), green(ellipse(-0.2, 0.7, 1.0, 0.3)), green(line ( -0.7, 3.7, -0.7, 0.7)), green(line ( 0.3, 3.7, 0.3, 0.7)), green(line ( -4.0, 1.5, -1.0, 1.5)), green(line ( -0.5, 4.5, -0.5, 4.0)), green(line ( 4.0, 0.0, 0.4, 1.2)) )}}} <B>Figure 3</B>. To the <B>Example 4</B> </TD> </TR> </TABLE> <H3>Example 6</H3>A pie, which has a cylindrical shape, is cut in 8 equal sectorial pieces along the radii. Find the volume of each piece if the diameter of the original pie is of 10 inches and its height is of 2 inches. <B>Solution</B> The volume of each piece is {{{1/8}}} of the volume of the entire pie, which is the cylinder of the diameter of 10 inches and the height of 2 inches. The volume of the entire pie is {{{V}}} = {{{pi}}}{{{r^2}}}{{{h}}} = {{{pi}}}*{{{5^2}}}*{{{2}}} = {{{50}}}{{{pi}}} cubic inches, and the volume of each piece is {{{1/8}}}{{{V}}} = {{{(50*3.14)/8}}} = 19.625 cubic inches. <B>Answer</B>. The volume of each piece of the pie is 19.625 cubic inches. My lessons on volume of cylinders 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> <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> Volume of cylinders <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>.
