Mathematics of Art

This problem can actually be solved with the brain in a few seconds:

Take a look at the amazing drawing I did since my skills at drawing are so great they are probably the greatest in the world

As you can see in the third panel, the shaded area is made up of six one-thirds of a circle. Therefore, the total area is the area of $2$ circles.

So the area is $2\pi r^{2} = 2\pi = \boxed{6.28...}$

Shaded area is composed of 6 big pieces + 6 small pieces. You can rotate one of the smaller pieces into the position shown in the picture. Then the shaded area is 1/3 of a circle. There are 6 of them. Thus total shaded area = 2 full circle's area.

We can clearly see $6$ "petals" (inner) and $6$ "funnels" (outer). Now, Hence, the total area is $\boxed{2\pi}$ .

The answer comes quick when simplified. Let us call the central circle "circle O, the seventh circle." The other six circles are the symmetrical "petals" of the flower-like figure. If one can calculate the shaded area added by each of the six unit circles remaining, then the total shaded area (call it A) equals six times the shaded area added by each of the six unit circles (call one U and its shaded area $S_U$ ).

The answer is most visible if one overlays a grid of equilateral triangles. If one uses a segment OU to divide its two adjacent secant arc areas, then it is clear that their positive area matches the two negative areas missing from the remaining shape. If those secant arcs are interchanged, the filled for the empties, then one is left with the filled shape occupying a one-third sweeping arc of the unit circle. $A = 6 \times S_U = 6 \times \frac{1}{3}\pi r^2$ $A=2\pi\approx 6.283$

For the inner circle, mark the thin orange slices as "a" and the white pieces as "b".

The single circle, with an area of pi, is 6a + 6b.

Look at the outer orange areas. These are actually made up of one a and two b's. If you total these up, you get 6a + 6a + 12b, which is two circles.

the inner circle gets cut by the outer six circles into six sections, so each of these outer orange areas cuts their respective circle on points that are 120 degrees apart. if these orange areas were straight edged arcs then they would each be 1/3 of the area of a circle. Each of the inner orange petals can fill in one of these orange areas to make a straight edge arc of a circle. There are six outer orange areas and six orange petals, so together they form six arcs of a circle where each arc is 1/3 of a circle. So all together all the orange areas total the area of two circles. Each circle has area one pi. So the answer is 2pi

Since my drawing skills are not as great as some others. I have rendered another's good hand drawn solution as images with MS Paint.

I will also admit that I first solved the problem "the hard way" as follows:

1. Identifying the 1/3 and 1/6 arc chords composed the unshaded areas as area removed from each unit circle.

2. Recalling how areas of sectors and chords can be calculated.

3. Working out value for one circle and adding in the inner petals, that all simplifies down to $2 \pi$

After entering that to earn the right to comment here, I saw the better answer and drew it up for presentation on my own timeline.