Lesson Solved problems on volume of cones

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<H2>Solved problems on volume of cones</H2>

In this lesson you will find typical solved problems on volume of cones.

The theoretical base for these problems is the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Volume/Volume-of-cones.lesson>Volume of cones</A>&nbsp; under the topic &nbsp;<B>Volume, metric volume</B>&nbsp; of the section &nbsp;<B>Geometry</B>&nbsp; in this site.

<H3>Problem 1</H3>Find the volume of a cone if the base radius of the cone is of &nbsp;4 cm&nbsp; and the height of the cone is of &nbsp;9 cm.

<B>Solution</B>

Apply the formula for the volume of a cone. &nbsp;The volume is

{{{V}}} = {{{1/3}}}{{{pi}}}.{{{r^2}}}{{{h}}} = {{{1/3}}}{{{pi}}}.{{{4^2}}}.{{{9}}} = {{{1/3}}}{{{pi}}}.{{{16)}}}.{{{9}}} = {{{48*pi}}} = 150.8 {{{cm^3}}} (approximately).

<B>Answer</B>. &nbsp;The volume of the cone is &nbsp;150.8 {{{cm^3}}} (approximately).

<H3>Problem 2</H3>Find the volume of a combposite solid body which comprises of two identical cones joined base to base &nbsp;(<B>Figure 3</B>), &nbsp;if their common base radius is of &nbsp;4 cm&nbsp; and the height

is of &nbsp;3 cm&nbsp; each.

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  <TD> 
<B>Solution</B>

We are given a 3D body comprised of two identical cones whose bases are joined and&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

overposed each to the other &nbsp;(<B>Figure 3</B>).

The volume of each single cone is

{{{V[1]}}} = {{{1/3}}}{{{pi}}}{{{r^2}}}{{{h}}} = {{{1/3}}}{{{pi}}}.{{{4^2}}}.{{{3}}} = {{{16}}}{{{pi}}} {{{cm^3}}}.

The volume of the given composite body is doubled this value, i.e.

{{{V}}} = {{{2*V[1]}}} = {{{32}}}{{{pi}}} = 3.14159*24 = 100.53 {{{cm^3}}} (approximately).

</TD>
  <TD>
{{{drawing( 210, 249,  -3.5, 3.5, -3.8, 4.5,

ellipse (0.0, 0.0, 6.0, 3.0),

line ( -2.74, 0.5,  0.0,  3.9),
            line (  2.80, 0.5,  0.0,  3.9),

line ( -2.80, -0.5,  0.0, -3.5),
            line (  2.85, -0.5,  0.0, -3.5)
)}}}

<B>Figure 3</B>. To the <B>Problem 2</B>

</TD>
 </TR>
</TABLE>

<B>Answer</B>. &nbsp;The volume of the given composite body is &nbsp;100.53 {{{cm^3}}} (approximately).

<H3>Problem 3</H3>A composite solid body comprises of the cone and the cylinder that have the same base radius measure. &nbsp;The cone and the cylinder are joined base to base in a way that the centers of their bases coincide &nbsp;(<B>Figure 4</B>). &nbsp;Find the volume of the given body if the common base radius is of &nbsp;4 cm&nbsp; and the height of the cone and the cylinder is of 3 cm.

<TABLE>
  <TR>
  <TD> 
<B>Solution</B>

We are given a 3D body comprised of the cone and the cylinder with identical base&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

radii measures whose bases are joined and overposed each to the other &nbsp;(<B>Figure 4</B>).

The volume of the cone is

{{{V[cone]}}} = {{{1/3}}}{{{pi}}}{{{r^2}}}{{{h}}} = {{{1/3}}}{{{pi}}}.{{{4^2}}}.{{{3}}} = {{{16*pi}}} = 50.265 {{{cm^3}}} (approximately).

The volume of the cylinder is

{{{V[cylinder]}}} = {{{pi}}}{{{r^2}}}{{{h}}} = {{{pi}}}.{{{4^2}}}.{{{3}}} = {{{48*pi}}} = 150.8 {{{cm^3}}} (approximately).

The total volume of the composite solid body is the sum of these values

{{{V}}} =  {{{V[cone]}}} + {{{V[cylinder]}}} = {{{1/3}}}{{{pi}}}{{{r^2}}}{{{h}}} +  {{{pi}}}{{{r^2}}}{{{h}}} = 50.266 + 150.8 = 201.06 {{{cm^3}}} (approximately).

</TD>
  <TD>
{{{drawing( 210, 300,  -3.5, 3.5, -5.5, 4.5,

ellipse (0.0, 0.0, 6.0, 2.0),

line ( -2.68, 0.5,  0.0,  3.9),
            line (  2.73, 0.5,  0.0,  3.9),

ellipse (0.0, -4.0, 6.0, 2.0),

line ( -3.0,  0.0, -3.0, -4.0),
            line (  3.0,  0.0,  3.0, -4.0)
)}}}

<B>Figure 4</B>. To the <B>Problem 3</B>

</TD>
 </TR>
</TABLE>

<B>Answer</B>. &nbsp;The volume of the composite body is &nbsp;201.06 {{{cm^3}}} (approximately).

<H3>Problem 4</H3>Find the volume of a body &nbsp;(a truncated cone)&nbsp; obtained from a cone with the base radius of &nbsp;4 cm&nbsp; and the height of &nbsp;6 cm&nbsp; after cutting off the part of the cone by the plane parallel to the base in a way that the cutting plane bisects the height of the original cone &nbsp;(<B>Figure 5</B>).

<TABLE>
  <TR>
  <TD> 
<B>Solution</B>

The strategy solving this problem is to find first the volume of the entire cone with&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

the base radius of &nbsp;6 cm&nbsp; and the height of &nbsp;8 cm&nbsp; and then to distract the volume

of the cone with the base radius of &nbsp;3 cm&nbsp; and the height of &nbsp;4 cm.

The volume of the original entire cone is &nbsp;&nbsp;{{{1/3}}}{{{pi*r^2*h}}}, &nbsp;where &nbsp;{{{r}}}= 4 cm&nbsp; is the cone base

radius and &nbsp;{{{h}}}= 6 cm&nbsp; is the cone height. &nbsp;So, the volume of the cone is

{{{V}}} = {{{1/3}}}{{{pi*4^2*6}}} = {{{32*pi}}} {{{cm^3}}}.

The small cone has the base radius of &nbsp;{{{1/2}}}{{{4}}} = 2 cm&nbsp; and the height of &nbsp;{{{1/2}}}{{{6}}} = 3 cm.

Therefore, &nbsp;the volume of the small cone is &nbsp;{{{1/3}}}{{{pi*2^2*3}}} = {{{4*pi}}} {{{cm^3}}}.

</TD>
  <TD> 
{{{drawing( 210, 249,  -3.5, 3.5, -3.8, 4.5,

ellipse (0.0, -2.0, 6.0, 3.0),

ellipse (0.0,  1.0, 3.0, 1.5),

line ( -2.75, -1.5,  -1.375,  1.25),
            line (  2.82, -1.5,   1.410,  1.25),

green(line ( -1.375,  1.25,  0.0,  4.0)),
      green(line (  1.410,  1.25,  0.0,  4.0)),

green(line (0, -2, 0, 4))
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure 5</B>. To the &nbsp;<B>Problem 4</B>

</TD>
 </TR>
</TABLE>

Thus the volume of the truncated cone under consideration is &nbsp;{{{32*pi}}} - {{{4*pi}}} = {{{28*pi}}} = 3.14159*28 = 87.96 {{{cm^3}}} (approximately).

<B>Answer</B>. &nbsp;The volume of the truncated cone is &nbsp;87.96 {{{cm^3}}} (approximately).

My lessons on volume of cones and other 3D solid bodies in this site are

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  <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>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

</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>

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 </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>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

</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>

Solved problems on volume of cones

<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>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

</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>

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</Table>

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.