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Radii of polyhedra

By Tom Button. Posted December 2, 2012

(At last month’s big MathsJam conference, we asked a few people who gave particularly interesting talks if they’d like to write something for the site. A surprising number said yes. First to arrive in the submissions pile was this piece by Tom Button.)

The formula for the surface area of a sphere, $A=4\pi r^2$, is the derivative of the formula for the volume of a sphere: $V=\frac{4}{3}\pi r^3$.

This result does not hold for a cube with side length $a$ if the surface area and volume are written in terms of $a$. However, if the surface area and volume are written in terms of half the side length, $r=\frac{a}{2}$, you get the surface area $A=24r^2$, which is the derivative of the volume, $V=8r^3$.