After reading the article on Poincare's conjecture in the Economist, I became curious about simplified 3-dimensional objects.

Excerpt:

Let's take a cube and simplify it into a circle. Could we then use equations ment for circles for the simplified shape, ie calculate the cube's surface area using S=pii*radius²?

How would the math look like for such calculations?

Whilst you could possibly work out how areas change under certain transformations, this is not what the poincare conjecture (or topology) is about. You're just doing some complicated sums (and undoubtedly integrals) to work out something quite trivial.

Topological onjects don't have area (hypervolume) in any well-defined sense. You can distort them in ways that change their volume without changing their topoogical properties.

but the idea of peremans proof, or hamiltons idea, was to impose a metric on the manifold, and show how to deforkm the metric until it became flat. manifolds with a metric do have "area" or volume, of course