Show that if a closed planar curve has perimeter zero then its enclosed area is zero

Question

Show that if $C$ is a planar closed curve of perimeter $P=0$, then it encloses an area of $A=0$.

This might seem like a stupid and obvious question, but the only mathematically sound proof I can think of is using the isoperimetric inequality, which would give us $$0^2=P^2 \geq 4\pi A,$$

thus obtaining $4\pi A = 0$ and so $A=0$ (since the area is, by definition, nonnegative).

Anybody have any simple or intuitive, rigorous proofs?

Answer 1

Pick two points $p,q$ on the curve. Then the arc from $p$ to $q$ has length $\ge |q-p|$. Conclude that $q=p$.

Answer 2

Given your tags, it's not clear what level of mathematical tools you have at your disposal. But a general strategy that could work at nearly any level would be to attack the contrapositive. That is, if the area of the region is not zero, then you should be able to choose some circle of non-zero radius that lies entirely in the area. Then it's just a matter of arguing that the perimeter of the entire area would have to be larger than the (non-zero) perimeter of that circle.