The trapezoid is trapped in the circle!

The trapezoid can be split into three congruent equilateral triangles as shown below: Since the radius, which is the same as the sides as the sides of the equilateral triangles, is $1$ , we can use our equilateral triangle area formula to find out the area: $A=\frac{s^2 \cdot \sqrt3}{4}$ Our side length is $1$ , so therefore we can get rid of the $s^2$ in the equation: $A=\frac{\sqrt3}{4}$ Since we have three equilateral triangles, remember to multiply by three: $A=\frac{3 \sqrt3}{4}$ That’s our area, so $a+b+c=\boxed{10}$

Consider my diagram. The trapezoid is composed of three congruent equilateral triangles. The area of an equilateral triangle is given by $A=\dfrac{\sqrt{3}}{4}x^2$ where $x$ is the side length. Therefore, the area of the trapezoid is $A=3\left(\dfrac{\sqrt{3}}{4}\right)(1^2)=\dfrac{3\sqrt{3}}{4}$ .

Thus, the desired answer is $a+b+c=3+3+4=\color{#69047E}\large{\boxed{10}}$