Area between three circles

Let the centers of the three circles be joined so as to form an equilateral triangle of side, $a$ .

Area of this triangle is given to be $A$ $=$ $\frac{\sqrt{3}}{4}$ $a^2$

The formation of such a triangle forms sectors in each circle. These sectors are alike since radii of the circles is the same i.e $(\frac{a}{2}$ )

The area of such sectors is given by $A$ $=$ $\frac{1}{2}$ $(0.5a)^2$ $\theta$

Area of the required region is simply given by

$\frac{\sqrt{3}}{4}$ $a^2$ $-$ $3$ $\frac{1}{2}$ $(0.5a)^2$ $\theta$

Now since it is an equilateral triangle $\theta$ $=$ $\frac{\pi}{3}$ and $0.5a$ $=$ $10cm$

Substituting these values we get the area of the region to be $\approx$ $16$ $cm^2$

Consider my diagram. Connect the centers of the three circles to form an equilateral triangle of side length $20$ . Observed that there are three sectors of a circle. If you add the areas of the three circular sectors, it would be the area of a semi-circle because the sum of the interior angles of a triangle is $180^\circ$ . So the area of the three circular sectors (yellow region is) $\dfrac{1}{2} \pi (10)^2=50 \pi$ . The area of an equilateral triangle is given by $\dfrac{\sqrt{3}}{4}x^2$ where $x$ is the side length. So the area of the equilateral triangle is $\dfrac{\sqrt{3}}{4}(20)^2=100\sqrt{3}$ . The area of the green part (shaded region in the problem) is equal to the area of the equilateral triangle minus the area of the yellow part. We have

$A=100\sqrt{3}-50\pi \approx \boxed{16.12544808}$