will be used to show that the formula for the volume of a sphere holds. For this purpose, consider a and a right with a removed from its interior, each with the same radius and height.
Now, consider a that cuts the at a height
x and is to the bases of the solids.
The of each will be calculated one at a time.
Finding the Hemisphere's Cross-Sectional Area
Draw a with height x, base y, and r. Here, x is the distance between the center of the base of the hemisphere and the center of the cross sectional , y is the radius of the cross sectional circle, and r is the radius of the hemisphere.
Using the , an for
y can be found.
Since
y is a distance, only the is considered. Therefore, the area
AH of the circular cross-section can be found using the formula for . For consistency,
y will be used in place of
r in the standard formula.
This equation gives the area of the cross-section of the hemisphere at altitude
x. Finding the Cylinder's Cross-Sectional Area
The area of the cross-section of the cylinder can be found similarly. The cross-section's area is equal to the area between two circles. Since the height and the radius of the cylinder are equal, an right triangle can be formed inside the cylinder. Therefore, the radius of the smaller circle is also
x.
Now that the radii of the circles are known, the area
AC of the cross-section can be calculated. It is the difference between the area
AG of the greater circle and the area
AS of the smaller circle.
AC=AG−AS
AC=πr2−πx2
AC=π(r2−x2)
The area of the cross-section of the cylinder at altitude
x can be found by using the above equation.
Conclusion
It can be stated that both solids have the same cross-sectional area at every altitude.
AH=π(r2−x2)=AC
Moreover, they have the same height. By Cavalieri's Principle, the hemisphere and the cylinder with a cone removed from its interior have the same volume.
If the is subtracted from the , the volume of the hemisphere can be found.
Vhemisphere=Vcylinder−Vcone
Vhemisphere=πr3−31πr3
Vhemisphere=33πr3−31πr3
Vhemisphere=33πr3−3πr3
Vhemisphere=32πr3
Finally, by multiplying the volume of the hemisphere by
2, the formula for the volume of a sphere will be obtained.
Vsphere=2⋅Vhemisphere=2⋅32πr3=34πr3