Two Circles, Apart For A Triangle

Geometry Level 2

If $r=1$ , then what is the area coloured in purple?

Details:

The blue sector is a perfect circle tangent to all three sides of the red equilateral triangle.
The red equilateral triangle is inscribed to the purple circle.

2\pi-\sqrt{3}

4\pi-3

4\pi-6\sqrt{3}

4\pi-3\sqrt{3}

2\pi-3\sqrt{3}

\pi-3\sqrt{3}