Area of Sacred Flower

Question

If each circle is unit circle then what will be area of given flower ?

My attempt : First I calculated comman area of two circles $a=2({1\over3}-{\sqrt3\over4})$

And area of one of 6 small leafs in centre $b={1\over3}-{\sqrt3\over2}$

I approach this question by drawing 6 outside circles one by one . And calculating area in each step

1. 1
2. 2-a
3. 3-2a+b
4. 4-3a+2b
5. 5-4a+3b
6. 6-5a+4b

Middle circle completely lies within 6 circle . So this is final answer ?

Is this solution correct or I did something wrong ? Is there any other unique and beautiful solutions by using probability or coordinate geometry ?

Answer 1

Here is another way: draw a circle of radius $2$ which has the same centre as the central circle (call this centre $O$) and touches all $6$ outer circles. Let it touch the two rightmost circles at $A$ and $B$, and let these circles have centres $P$ and $Q$, and their other intersection point be $B$.

Then the required area is the area of the larger circle minus $6$ times the area between the outer circle and the two rightmost circles. This area is the area of sector $OAB$ minus $2$ times sector $APB$ minus the area of the rhombus $OPQR$.

So the area is $$4\pi-6\left(\frac{2\pi}{3}-\frac{\pi}{3}-\frac{\sqrt{3}}{2}\right)$$ $$=2\pi+3\sqrt{3}$$