This world needs more coin of such shape!

$\angle A_1OA_2 = \frac{2\pi}{7}$

$\angle A_1A_5A_2 = \frac {\pi}{7}$

$\angle A_1OA_2 = \frac {\pi}{14}$

$\angle A_1OA_5 = \frac {6\pi}{7}$

Area triangle $A_1OA_5 = \frac {1}{2} \sin \frac {6\pi}{7}$

Length $A_1A_5 = \frac {\sin \frac {6\pi}{7}}{\sin \frac{\pi}{14}}$

Area sector $A_1A_5A_2$ = $\frac {1}{2} \angle A_1A_5A_2 \times (A_1A_5)^2$

Area bounded by radii $OA_1, OA_2$ and arclength $A_1A_2$ = Area sector $A_1A_5A_2$ - 2 Area triangle $A_1OA_5$

= $\frac{\pi}{14}$ $\sin^2\frac{6\pi}{7} \div \sin^2\frac{\pi}{14}$ - $\sin\frac{6\pi}{7}$

Area Reuleaux heptagon is 7 $\times$ this =2.93488

Check: upper bound $\pi$ , Lower bound $\frac {7}{2}\sin \frac{2\pi}{7}$ =2.73641

Similarly for odd Reuleaux n-gon

Area n-gon
= $\frac{\pi}{2n}\sin^2 \frac{(n-1)\pi}{n} \div \sin^2\frac{\pi}{2n} - \sin\frac{(n-1)\pi}{n}$

Sorry about the clunkiness of the typesetting ; I'm a 46 year old LaTeX virgin (no sniggering at the back there). Next time I'll just upload a photo.

Area of Reuleaux $n$ -gon, when $n$ is an odd positive integer, when $R$ is the circumradius of associated regular $n$ -gon,

$\frac{1}{2}nR^2 \left( \sin{\frac{2\pi}{n}} + 4\left( \frac{\pi}{2n} - \sin{\frac{\pi}{2n}} \right) \times \cos^2{\frac{\pi}{2n}} \right)$

Further algebraic simplification may be possible.