How
to find the volume of a sphere


1. What you should know:


2. Study the two
figures below.
Figure 1 shows a hemisphere
and we like to find its volume. We like to find
the crosssectional area of a thin layer with a vertical distance a from the center of the base. Using the
Pythagoras theorem, the square of the radius of the
crosssection (in red) is _{} Hence the
crosssectional area, which is a circle is _{} Figure 2 shows
a cylinder with height r, radius r, with an inverted cone inside. We like to
find the volume of the cylinder which is outside the cone (yellow portion). The
crosssectional area of a thin layer with a vertical distance a (same as figure 1) from the center of the base
consists of two concentric circles. We like to find the area which is outside the small circle but inside the large circle. (shown in red) Please note
that the radius and height of the cone are both equal to r. With a little
calculation, the radii of the two concentric circles of the crosssection are
r and a,
with r being the radius of the bigger circle. Hence the
crosssectional area is _{}. As a result the
crosssectional areas of both figure 1 and figure 2 are the same, both are
equal to _{}. 

3. As the cross section areas are the same and the height of the whole solids are the same, that is, _{} and r respectively, they have the same volume. Therefore: Volume of hemisphere = Volume of cylinder – volume of
inverted cone _{} _{} \Volume of a sphere = 2 x
volume of hemisphere _{} (It is noted
that the crosssectional areas of the solids in both figures may change with
different heights from the center of the base. However, this does not affect
our proof.) 