A shape is called equable, I am told, if the numerical value of the circumference and the area (2D) or of the surface and volume (3D) are equal. How silly, I thought at first, as this depends on the measuring unit, so I could for any given shape invent a measuring unit that would make it equable.

On second thoughts, however, there are some interesting aspects to this concept (if we stick with one measuring unit, eg cm). Take a square for example -- to make it equable it would have to have a side length of 4cm, making both area and circumference 16. An equable circle would have a radius of 2cm. Notice something curious about those two shapes ?

The thing is that the circle fits exactly inside the square. Moreover, I predict that every polygon that wraps around a circle with r=2cm (all sides must be tangents) will be equable. (So the equable circle emerges as the n=infinity case of a whole series of equable polygons.) And I can prove it.

If that's too easy, prove the analogous statement for spheres with r=3cm.

If that's still too easy, try 4 dimensions and hyperspheres with r=4cm.

And I'm sure there must be a general proof for n dimensions ... :)

enjoy !