How
to find the volume of a sphere
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1. What you should know:
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2. Study the two
figures below.
Figure 1 shows a hemisphere
and we like to find its volume. We like to find
the cross-sectional 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
cross-section (in red) is Hence the
cross-sectional 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
cross-sectional 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 cross-section are
r and a,
with r being the radius of the bigger circle. Hence the
cross-sectional area is As a result the
cross-sectional areas of both figure 1 and figure 2 are the same, both are
equal to |
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3. As the cross section areas are the same and the height of
the whole solids are the same, that is,
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 cross-sectional 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.) |
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