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 \(A_1 A_2 \) is drawn from center \(A_5 \), arc \(A_2 A_3 \) is drawn from center \(A_6\), arc \(A_3 A_4 \) is drawn from center \(A_7\), and so on.

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

Bonus: Generalize this for all Reuleaux \(n\)-gon where \(n\) is an odd number.

Inspiration

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