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First I notice that the distance between the tips of the petals is one radius, let's call the radius r.

If we use x to denote the height of the triangle formed by that line and the two radii connecting it to the center, then by the Pythagorean theorem we get:

x² + r²/4 = r²

x² = 3 × r²/4

x = (r/2) × √3

Let A_s be the area of the sector and A_t be the area of the triangle:

A_s = r² × π/6

A_t = x × r/2

We put in our expression for x into this and get:

A_t = (r/2) × (√3) × r/2

A_t = (r²/4) × √3

Let A_h be the area of half a petal. And note that half a petal is also found between the triangle and the circumference of the circle. Therefore:

A_h = A_s - A_t

A_h = r² × π/6 - (r²/4) × √3

A_h = r² × (π/6 - (1/4) × √3)

We have 12 half petals, so:

A_flower = 12 × A_h

A_flower = 12 × r² × (π/6 - (1/4) × √3)

A_flower = r² × (2 × π - 3 × √3)

A_flower = r² × (2 × π - √27)