I was asked to explain why the volume of a sphere is $\frac{4}{3}\pi r^3$ to a student that does not have the knowledge of calculus. In doing so I thought of an argument and I cannot seem to find that argument elsewhere so far. The proofs for the area of the sphere that I know of are:
i) Integrating up spherical shells or direct integration from spherical polar coordinates etc
ii) Cavalieri's Principle
iii) Archimedes proof.
All of which can be found here: https://proofwiki.org/wiki/Volume_of_Sphere
Are there any other proofs out there ? How can I go about finding them ? (Maybe proofs that assume some properties of the sphere, such as its surface area...)
One visualization of the area of a circle of radius $r$ is to approximate it by a large number of triangles of height $r$ each having a vertex at the center of the circle and the opposite edge on the circumference. The total of the triangle base edge lengths is $b=2\pi r$ and since the height is $h=r$ the total area is $\frac12hb=\pi r^2.$
If you show that the sphere has area $A=4\pi r^2$ then a three dimensional visualization with pyramids of height $h=r$ and bases on the sphere gives you a volume of $\frac13hA=\frac43\pi r^3.$
A potential disadvantage of this approach is that it may be harder to intuitively grasp the area of the sphere than to arrive at the volume by other means.