I have a loop of string whose total length is $10$. I place it around a unit disk and pull a point on the string away from the disk until the string is taut, as shown below.
I drag this point around the disk in all directions, always keeping the string taut, tracing out a loop. What is the area inside this resulting loop?
For sake of exposition, let's assume that the unit circle is centered at the origin, $O = (0,0),$ and that I am pulling the string taut along the $x$-axis until it reaches the point $P = (r,0)$. Let's first determine the value of $r.$
If the string is taut, then the string will form a straight line that is tangent to the circle at say some point $Q$. If we draw a radius from $O$ to $Q$, then $\Delta OQP$ is a right triangle, where $\angle OQP$ is the right angle. Since the hypotenuse has length $r$ and the radius $OQ$ has unit length, we see that $|PQ| = \sqrt{r^2-1}$ from some guy named Pythagoras. Additionally we see that $m \angle QOP = \tan^{-1} \sqrt{r^2-1}.$ Therefore, if we try to calculate the total length of the string, we get $$\ell (r) = 2\pi + 2 \sqrt{r^2-1} - 2 \tan^{-1} \sqrt{r^2-1},$$ which is an increasing function from $\ell: [1,\infty) \to [2\pi, \infty).$ Since $\ell$ is monotonic it can be inverted, so that if we know that the string has length $10$, then we get $$\hat{r} = \ell^{-1} (10) \approx 3.2753266928\dots.$$
Since the choice of stretching the string along the $x$-axis was arbitrary, we see that as we drag the point around the disk in all directions, the maximum extent from the origin will constantly be $\hat{r},$ so the locus traces out a circle with radius $\hat{r},$ which has area $$A = \pi \hat{r}^2 \approx 33.7022675393\dots.$$
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