This world needs more coin of such shape!

Geometry Level 5

The diagram above shows a Reuleaux septagon drawn from a regular septagon with circumradius 1, where arc A1A2A_1 A_2 is drawn from center A5A_5 , arc A2A3A_2 A_3 is drawn from center A6A_6, arc A3A4A_3 A_4 is drawn from center A7A_7, and so on.

Find the area of the Reuleaux septagon to 5 decimal places.

Bonus: Generalize this for all Reuleaux nn-gon where nn is an odd number.


Inspiration

You may want to read up Reuleaux triangle first.
Image Credit: Wikimedia Reuleaux polygons by LEMeZza
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