Is this obvious? Approximation of pi?

[apple2357]

At first glance looks fine, but I think that it's not obvious (though it's true) that the circle circumference is smaller than the outer hexagon?

 

[blamocur]

Good point! One can show that both areas are half-products of the radius by the respective perimeters.

 

[Dr.Peterson]

I imagine this exercise is modeled on Archimedes' approximation of pi; but he used area, not perimeter, and the former leads to an obvious inequality, while the latter, while true, is not obvious. It's easy to confuse the two, and think that both are sufficient for a proof.

 

[apple2357]