When Archimedes found the upper and lower boundary for the value of pi, he used an $\color{red}{\textrm{inscribed regular polygon}}$ and a $\color{blue}{\textrm{encapsulating outer regular polygon}}$. He then let the number of edges increase, as depicted here:
I would love to know if it was proven or known at the time, that the $\color{red}{\textrm{inscribed regular polygon}}$ had a smaller perimeter than the $\color{green}{\textrm{circle}}$? I am not a mathematician but I feel like this should be proven before continuing finding the bounds of pi.
I would arguer the outer object, has to have a greater perimeter, but the inner object could be constructed not to be smaller but actually have a greater perimeter, like this star-like-shape:
I would love to hear your input on this, and if you have any idea how to prove an inscribed regular polygon has to have a perimeter smaller than that of a outer encapsulating circle?