Question

What structure encloses the most space with the least surface area (has the the highest volume to surface area ratio)?

Answer 1

That structure would be the Sphere.

With a Volume of V = 4/3*π*r**3

And a surface area of A = 4*π*r**2

r =(A/(4*π))**1/2

V = 4/3*π* [A^(3/2)/(8 π^(3/2))] = A^(3/2)/(6 sqrt(π))

We can graph V in tems of A

And compare it to the graph y = x

We can see that the Volume increases more rapidly compared to the Area