1. **Problem statement:** We have two circles with radii $a$ and $b$ where $a > b$, and the distance between their centers is $c$. We want to find the area of their overlap. 2. **Key idea:** The overlap area of two circles can be found by summing the areas of two circular segments formed by the chord of intersection. 3. **Using the cosine rule:** Consider the triangle formed by the two circle centers and one of the intersection points. The sides are $a$, $b$, and $c$. The angle opposite side $c$ in the triangle with sides $a$, $b$, and $c$ is found using the cosine rule: $$\cos \theta_a = \frac{a^2 + c^2 - b^2}{2ac}$$ Similarly, the angle opposite side $c$ in the triangle with sides $b$, $a$, and $c$ is: $$\cos \theta_b = \frac{b^2 + c^2 - a^2}{2bc}$$ 4. **Calculate the segment areas:** The area of the circular segment for radius $a$ is: $$A_a = a^2 \arccos\left(\frac{c^2 + a^2 - b^2}{2ac}\right) - \frac{1}{2} \sqrt{(-a + b + c)(a - b + c)(a + b - c)(a + b + c)}$$ Similarly, for radius $b$: $$A_b = b^2 \arccos\left(\frac{c^2 + b^2 - a^2}{2bc}\right) - \frac{1}{2} \sqrt{(-a + b + c)(a - b + c)(a + b - c)(a + b + c)}$$ 5. **Total overlap area:** The total overlap area is the sum of these two segment areas: $$\text{Overlap area} = A_a + A_b$$ 6. **Summary:** The cosine rule is used on the triangle formed by the centers of the two circles and one intersection point to find the angles needed for the segment areas. The chord length and the segment heights come from these angles. This formula assumes $|a-b| < c < a+b$ so the circles intersect. **Final formula:** $$\text{Overlap area} = a^2 \arccos\left(\frac{c^2 + a^2 - b^2}{2ac}\right) + b^2 \arccos\left(\frac{c^2 + b^2 - a^2}{2bc}\right) - \frac{1}{2} \sqrt{(-a + b + c)(a - b + c)(a + b - c)(a + b + c)}$$