Volume of a Sphere, Cavalieri's Principle, Cups

####Making Sense of the Volume of a Sphere: Have a Cup of Math, Please.
#####Three different cups with a volume of approximately 2 fl oz.

The volume of a sphere of radius R is (4/3) πR^3. But why? One can certainly use a bit of calculus to find the volume of a sphere in secondary and college mathematics. In school mathematics, however, how can we make sense of the formula and, specifically, where does that (4/3) come from?

The present design allows one to explain the volume of a sphere using Cavalieri’s Principle. Specifically, it allows a semi-sphere of radius R to be aligned with a cylinder-minus-cone whose terminal radius and height are both R. When positioned appropriately (as shown in the figures), the cross-sections of the semi-sphere and the cylinder-minus-cone has the same area at any point of their height. This fact can be easily proved using the Pythagorean Theorem and simple comparison. Let’s start with the assumption that the volume of the cone of radius R and height R is (1/3) πR^3, which itself can be justified using Cavalieri’s Principle using a square pyramid as a reference (https://www.thingiverse.com/thing:3093547). Then, the cylinder-minus-cone has a volume of (2/3) πR^3, which is the same as that of the semi-sphere. Therefore, the volume of the whole sphere is (4/3)πR^3.

The cone can be taken from the cylinder in two ways: an upright cone and a tilted cone. Thus, there are two versions for the cylinder-minus-cone cup. In designing the cups, attention is paid to the inside of the cups so that liquid can be used to compare the volume (capacity) of both cups. The inner diameter is about 61mm, yielding a volume of approximately 2 fl oz.

####References
1. Cavalieri’s Principle. https://en.wikipedia.org/wiki/Cavalieri%27s_principle
2. https://www.britannica.com/science/Cavalieris-principle