Finding the area

Joining the centers of the circles forms an equilateral triangle with side length of 5. The shaded region is composed of one sector of a circle and two segments of a circle. Adding the areas of the three sectors gives us the area of the shaded region plus twice the area of the triangle.

The area of the triangle is $\dfrac{s^2\sqrt{3}}{4}=\dfrac{5^2\sqrt{3}}{4}=\dfrac{25\sqrt{3}}{4}$ .

The area of the three sectors is equivalent to the area of a semicircle, which is equal to $\dfrac{1}{2}\pi r^2=\dfrac{1}{2}\pi (25)=\dfrac{25}{2}\pi$ .

The area of the shaded region therefore is equal to the area of the three sectors minus twice the area of the triangle.

$\text{area of the shaded region}=\dfrac{25}{2}\pi-2(\dfrac{25\sqrt{3}}{4})=\dfrac{25}{2}\pi-\dfrac{25}{2}\sqrt{3}$

Finally,

$a+b+c=25+2+3=$ $\boxed{\color{#69047E}30}$