Proofs of the Volume of a Sphere.

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...)

• It might be useful to see your idea too. – David K Sep 2 '20 at 19:22

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 $$\frac13hA=\frac43\pi r^3.$$