Unit diameter pentagons with maximum area

Michael J. Mossinghoff, "A \$1 Problem." Amer. Math. Monthly 113 (2006), 385–402; jstor. I quote:

The isodiametric problems for polygons were first studied by Karl Reinhardt, Bieberbach's first student, in 1922 [22]. He solved the area problem for odd values of $n$, showing that the regular $n$-gon is best possible. Then, in an appendix that seems to have been missed in some of the later literature, he proved that the regular $n$-gon is never optimal when $n$ is an even number and $n\ge 6$.

Reinhardt's proof for the case of $n$ odd, and his example for $n$ even, is given in the article; it also states and cites the solutions for the cases $n=6$ (Bieri 1961, Graham 1975) and $n=8$ (Audet et al. 2002), and some information about the asymptotics as $n\to\infty$ (another Mossinghoff paper).

Just to add something concrete, here's a straightforward proof that $\lim_{n\to\infty} A(n) = \frac\pi4$. Let $K$ be any set with diameter at most 1; then $K-K\subseteq C$, where $C$ is the unit circle, so by the Brunn–Minkowski inequality, $\text{Area}(K) \le \frac14 \text{Area}(K-K) \le \frac14 \text{Area}(C) = \frac\pi4$. On the other hand, a regular $n$-gon with circumradius $\frac12$ has diameter at most 1 and contains a circle of radius $\frac12 \cos\frac\pi n$, and so $A(n)\ge\frac\pi4 \cos^2\frac\pi n$.

Circumscribe the polygon by a circle.

If n is even,

The longest diagonals pass through the center of the circle.Therefore $r=\frac{1}{2}$. $$\text{Area of the polygon =} \frac{1}{2}nr^2 \sin\frac{2\pi}{n}=\frac{1}{8}n \sin\frac{2\pi}{n}$$

If n is odd,

The longest diagonals will not pass through the center of the circle,angle subtended by the diagonal at the center = $\frac{n-1}{2}\frac{2\pi}{n}=\left(1-\frac{1}{n}\right)\pi$

Therefore, $2r\sin\frac{(n-1)\pi}{2n}=1\rightarrow r = \large\frac{1}{2\sin\frac{(n-1)\pi}{2n}}$ $$\text{Area of the polygon =} \frac{1}{2}nr^2 \sin\frac{2\pi}{n}=\frac{1}{8}n \sin\frac{2\pi}{n}\csc^2\frac{(n-1)\pi}{2n}$$

The red line shows the area in cases when n is even and the green line shows the odd cases.

As expected there is a very small difference between the 2 lines and they converge to $\pi/4$.It can also be observed that it is not absolutely monotonous.There are certain exceptions.