### How to find the volume of a sphere

1.  What you should know:

# Volume of cylinder # Volume of cone # Volume

#### Cross-sectional area x height

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 .

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 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.)