Lesson OVERVIEW of LESSONS on Volume of SPHERES

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<H2>OVERVIEW of LESSONS on Volume of Spheres</H2>For your convenience, &nbsp;this file contains

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- the list of my lessons on volume of spheres in this site,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- the formula for calculating the volume of spheres, &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- the list of relevant solved problems.

<H3>Spheres</H3>A &nbsp;<B>sphere</B>&nbsp; is a surface in a &nbsp;<B>3D</B>&nbsp; space all points of which are &nbsp;<B>equidistant</B>&nbsp; from one point called the &nbsp;<B>center of the sphere</B>.

<B>Figure 1a</B>&nbsp; shows the sphere.

A &nbsp;<B>radius of a sphere</B>&nbsp; is a straight segment connecting the center of the sphere with a point on the sphere surface &nbsp;(<B>Figure 1b</B>).

A &nbsp;<B>diameter of a sphere</B>&nbsp; is the straight segment passing through the center of the sphere and connecting the opposite points of the sphere  &nbsp;(<B>Figure 1c</B>).

&nbsp;&nbsp;{{{drawing( 180, 180,  -4.5, 4.5, -4.5, 4.5, 
                        circle( 0.0, 0.0, 4.0, 4.0),
                        circle( 0.0, 0.0, 0.1, 0.1),
                        locate(-0.2, -0.1, O),
                  green(ellipse(0.0, 0.0, 8, 4)) 
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure 1a</B>. A sphere
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

</TD>
  <TD>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 180, 180,  -4.5, 4.5, -4.5, 4.5, 
                        circle(0.0, 0.0, 4, 4),
                        circle( 0.0, 0.0, 0.1, 0.1),
                        locate(-0.2, -0.1, O),
                  green(ellipse( 0.0, 0.0, 8, 4)),
                  green(line( 0.0, 0.0, 2.4, 1.5)),
                        locate( 1.2, 0.9, r)
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure 1b</B>. A sphere 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and its radius

</TD>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 180, 180,  -4.5, 4.5, -4.5, 4.5, 
                        circle(0.0, 0.0, 4, 4),
                        circle( 0.0, 0.0, 0.1, 0.1),
                        locate(-0.2, -0.1, O),
                  green(ellipse( 0.0, 0.0, 8, 4)),
                  green(line(-2.4, -1.5, 2.4, 1.5)),
                        locate( 1.2, 0.9, d)
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure 1c</B>. A sphere
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and its diameter

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

<H3>Properties of spheres</H3>

1. &nbsp;Every section of a sphere by a plane is a circle.

2. &nbsp;A tangent segment to a sphere released from a point in &nbsp;<B>3D</B>&nbsp; space outside the sphere is perpendicular to the radius of the sphere drawn from its center

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;to the tangent point.

3. &nbsp;All tangent segments to a sphere released from one fixed point outside the sphere have the same length.

<H3>Formula for calculating the volume of spheres</H3>

The volume of a sphere</B>&nbsp; is &nbsp;{{{S}}} = &nbsp;{{{4/3}}}{{{pi}}}{{{r^3}}} = {{{1/6}}}{{{pi}}}{{{d^3}}},

where &nbsp;{{{r}}}&nbsp; is the radius of the sphere and &nbsp;{{{d}}}&nbsp; is the diameter of the sphere.

<H3>My lessons on volume of spheres in this site</H3>&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Volume/Volume-of-spheres.lesson>Volume of spheres</A>

under the topic &nbsp;<B>Volume, metric volume</B>&nbsp; of the section &nbsp;<B>Geometry</B>, &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-volume-of-spheres.lesson>Solved problems on volume of spheres</A>

under the topic &nbsp;<B>Geometry</B>&nbsp; of the section &nbsp;<B>Word problems</B>.

<H3>Solved problems on volume of spheres</H3>&nbsp;&nbsp;&nbsp;&nbsp;- Find the volume of a sphere if its radius is of &nbsp;10 cm.

&nbsp;&nbsp;&nbsp;&nbsp;- Find the volume of a sphere if its radius is of &nbsp;3 cm.

<TABLE>
  <TR>
  <TD>&nbsp;&nbsp;&nbsp;&nbsp;- Find the volume of a composite body comprised of a right circular cylinder and a hemisphere attached center-to-center&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;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;to one of the cylinder bases if both the cylinder diameter and the hemisphere diameter are of &nbsp;20 cm, &nbsp;and the cylinder

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;height is of &nbsp;40 cm.

&nbsp;&nbsp;&nbsp;&nbsp;- Find the volume of a composite body comprised of a right circular cylinder and a hemisphere attached center-to-center

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;to one of the cylinder bases if both the cylinder diameter and the hemisphere diameter are of &nbsp;10 cm, &nbsp;and the cylinder

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;height is of &nbsp;20 cm.

</TD>
  <TD>{{{drawing( 105, 98,  -3.5, 3.5, -1.0, 5.5,

ellipse( 0.5, 3.5, 3.0, 1.0),
            ellipse( 0.5, 0.0, 3.0, 1.0),

line( -1, 3.5, -1, 0.0),
              line(  2, 3.5,  2, 0.0),

arc ( 0.5, 3.5, 3.00, 3.00, 180, 360),
              arc ( 0.5, 3.5, 3.06, 3.06, 180, 360)
)}}}

</TD>
 </TR>
</TABLE>
<TABLE>
  <TR>
  <TD>&nbsp;&nbsp;&nbsp;&nbsp;- Find the volume of a composite body comprised of a cone and a hemisphere attached center-to-center to the cone 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;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if both the cone base diameter and the hemisphere diameter are of &nbsp;20 cm&nbsp; and the cone height is of &nbsp;10 cm.

&nbsp;&nbsp;&nbsp;&nbsp;- Find the volume of a composite body comprised of a cone and a hemisphere attached center-to-center to the cone base

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if both the cone base diameter and the hemisphere diameter are of &nbsp;10 cm&nbsp; and the cone height is of &nbsp;5 cm.

</TD>
  <TD>{{{drawing( 98, 83,  -3.0, 3.5, 0.0, 5.5,

ellipse( 0.5, 3.5, 3.0, 1.0),

line( -1.0, 3.5,  0.5, 0.0),
              line(  2.0, 3.5,  0.5, 0.0),

arc ( 0.5, 3.5, 3.00, 3.00, 180, 360),
              arc ( 0.5, 3.5, 3.06, 3.06, 180, 360)
)}}}
 </TD>
 </TR>
</TABLE><TABLE>
  <TR>
  <TD>&nbsp;&nbsp;&nbsp;&nbsp;- Find the volume of a composite body comprised of a cube and a hemisphere attached center-to-center to one of the&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;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;cube faces if both the cube edge measure and the hemisphere diameter are of &nbsp;20 cm.

&nbsp;&nbsp;&nbsp;&nbsp;- Find the volume of a composite body comprised of a cube and a hemisphere attached center-to-center to one of the

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;cube faces if both the cube edge measure and the hemisphere diameter are of &nbsp;10 cm.

</TD>
  <TD>
{{{drawing( 90, 90,  -2.5, 3.5, -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)),

ellipse( 0.52, 3.43, 2.54, 0.77),
            arc    ( 0.52, 3.43, 2.54, 2.54, 175, 354)
)}}}
 </TD>
 </TR>
</TABLE><TABLE>
  <TR>
  <TD>&nbsp;&nbsp;&nbsp;&nbsp;- Find the volume of the sphere inscribed in a cone if the base diameter of the cone is of &nbsp;24 cm&nbsp; and the height of the&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;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;cone is of &nbsp;16 cm.

&nbsp;&nbsp;&nbsp;&nbsp;- Find the volume of the sphere inscribed in a cone if the base diameter of the cone is of &nbsp;30 cm&nbsp; and the height of the

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;cone is of &nbsp;36 cm.

</TD>
  <TD>
{{{drawing( 105, 124,  -3.5, 3.5, -3.8, 4.5,

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

green(ellipse (0.0,  0.8, 3.2, 1.6)),

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

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

green(line (0, -2, 0, 4)),
            circle(  0.0, -0.12, 1.88, 1.88)
)}}}

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

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

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  <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>&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>

</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>&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>

<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>&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>

Overview of lessons on volume of spheres

</TD>
 </TR>
</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>.