Volume of spheres (Cavalieri´s principle)
Calculating the volume of a sphere is possible using an appropriate cylinder and cone.
volume of sphere, Cavalieri´s principle, calculating volume, solids, sphere, mathematics
Demonstration of Cavalieri´s principle
Take two solids and place them on one plane. Intersect them with two planes parallel to their base and examine the two solids and their cross-sections according to the following properties:
- The areas of their bases are equal.
- The areas of all cross sections parallel to their bases are equal.
- The heights of the two solids are equal.
If all of these are true, the volumes of the two solids are equal.
Cavalieri´s principle helps a lot in calculating the volume of spheres. Without it, higher mathematical methods would need to be applied in order to come to a result.
Let´s consider a hemisphere of radius r with its cross-section, and a cylinder which lies in the same plane. The radius of the circular base and the height of the cylinder are r. Let´s cut from the cylinder an upturned cone, with both radius and height r. In the animation, these solids are shown together with their mirror images relative to the plane. The areas of the bases of the two solids are equal.
When examining their cross-sections parallel to this plane we have to calculate the area of the cross-sections, which is at a height h.
In the case of the sphere the cross-section is a circle. Due to the Pythagorean theorem, the square of the circle's radius equals to r² - h² , thus its area is
In the case of the other solid the cross-section is an annulus with an external radius r and an internal radius h. Its area is
In other words, given two solids, the surface areas of their cross-sections parallel to their base are equal.
Due to the formation of the two solids, the heights of the two solids are equal.
All the conditions of Cavalieri´s principle are met, therefore the two solids have an equal volume.
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