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...)

  • $\begingroup$ It might be useful to see your idea too. $\endgroup$ – 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 $\frac13hA=\frac43\pi r^3.$

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.

  • $\begingroup$ Thank you, that's what I thought of as well, do you happen to have a link or source or something where this is done ? ( like dealing with the issue of convergence at the cap etc ?) $\endgroup$ – Rohan Nuckchady Sep 2 '20 at 19:42
  • $\begingroup$ For the pyramids you just need to subdivide the sphere's surface into small regions. They don't have to be regular, just cover the surface. You could start with a soccer ball and then put grids in the hexagons and pentagons. Or use latitude and longitude, no problem, pyramids near the pole might just have very skinny bases. $\endgroup$ – David K Sep 2 '20 at 19:47
  • $\begingroup$ Yeah, and to prove that as we make the pyramids smaller, the combined volume of the pyramids converges to the volume of the sphere, it seems sufficient to 'extend the pyramids to the outside of the sphere till their base is tangential to the sphere' and use some sort of sandwiching argument ? $\endgroup$ – Rohan Nuckchady Sep 2 '20 at 19:52
  • $\begingroup$ Yes, that's how I would make the argument more rigorous. $\endgroup$ – David K Sep 2 '20 at 19:57

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