1. **Problem statement:** We have two circles with radii $a$ and $b$, and the distance between their centers is $c$. We want to find the area of their overlapping region. 2. **Formula and explanation:** The area of overlap (intersection) of two circles can be found using the formula: $$ A = a^2 \cos^{-1}\left(\frac{c^2 + a^2 - b^2}{2ac}\right) + b^2 \cos^{-1}\left(\frac{c^2 + b^2 - a^2}{2bc}\right) - \frac{1}{2} \sqrt{(-c+a+b)(c+a-b)(c - a + b)(c + a + b)} $$ This formula comes from summing the areas of the circular segments formed by the chord of intersection. 3. **Important rules:** - The distance $c$ must satisfy $|a-b| \leq c \leq a+b$ for the circles to overlap. - If $c \geq a+b$, the circles do not overlap (area = 0). - If $c \leq |a-b|$, one circle is completely inside the other (area = area of smaller circle). 4. **Intermediate work:** - Calculate the angles for the circular segments: $$\theta_a = \cos^{-1}\left(\frac{c^2 + a^2 - b^2}{2ac}\right)$$ $$\theta_b = \cos^{-1}\left(\frac{c^2 + b^2 - a^2}{2bc}\right)$$ - Calculate the area of the triangular part: $$T = \frac{1}{2} \sqrt{(-c+a+b)(c+a-b)(c - a + b)(c + a + b)}$$ 5. **Final answer:** $$ \boxed{A = a^2 \theta_a + b^2 \theta_b - T} $$ This gives the area of the overlapping region of the two circles.