We have two convex sets $S,T$ such that $S\subseteq T$. Prove or disprove that the circumference of $S$ must be smaller than that of $T$.

Question

This question originated from an exercise asking to compare the arclength of $y=x^2$ between $(0,0)$ and $(1,1)$ and $\frac{\pi}{2}$.

The solution starts by constructing the circle with center $(0,1)$ and radius $1$, and showing that the arc of parabola sits entirely within the circle, thus the arclength is less than $\frac{\pi}{2}$.

However, I'm not entirely sure that this is a valid proof, for the last step invokes some geometric intuition and I do not see an obvious way to rephrase it rigorously.