Considering a Sphere

4 hours ago

The author explores visualizing the volume and surface area of a sphere, starting with analogies from a circle (where circumference is the derivative of area) and a cube.
Using Cavalieri's principle and geometric transformations, it's shown that a square-based pyramid with base area 1 and height 1 has volume 1/3, which scales to general formulas.
The volume of a sphere is derived as 1/3 times the radius times the surface area, with the surface area given by the Archimedes Hat Box Theorem as 4πr², leading to the volume formula 4/3πr³.
An 'egg-shell' argument is applied: increasing the radius by dr increases volume by 4πr²·dr, integrating from 0 to r gives the volume, and this method is also verified for a cube.
The post concludes with acknowledgments to contributors and a note on extending the technique to other shapes like a tetrahedron.

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