Lune Area 46737D

1. **Problem Statement:** Find the area of the lune formed by two intersecting circles with radii $r$ and $R$, where $R > r$. 2. **Understanding the lune:** The lune is the crescent-shaped region bounded by the arcs of the two circles. 3. **Formula and approach:** The area of the lune can be found by subtracting the area of the smaller circular segment from the area of the larger circular segment formed by the intersection. 4. **Key formulas:** - Area of a circular segment with radius $a$ and chord length $c$ or central angle $\theta$ is: $$\text{Segment area} = \frac{a^2}{2}(\theta - \sin\theta)$$ where $\theta$ is in radians. 5. **Steps to find the lune area:** - Find the distance $d$ between the centers of the two circles. - Calculate the central angles $\theta_R$ and $\theta_r$ for the larger and smaller circles respectively using the law of cosines: $$\theta_R = 2\cos^{-1}\left(\frac{d^2 + R^2 - r^2}{2 d R}\right)$$ $$\theta_r = 2\cos^{-1}\left(\frac{d^2 + r^2 - R^2}{2 d r}\right)$$ - Calculate the areas of the segments: $$A_R = \frac{R^2}{2}(\theta_R - \sin\theta_R)$$ $$A_r = \frac{r^2}{2}(\theta_r - \sin\theta_r)$$ 6. **Final lune area:** $$\text{Area of lune} = A_R - A_r = \frac{R^2}{2}(\theta_R - \sin\theta_R) - \frac{r^2}{2}(\theta_r - \sin\theta_r)$$ 7. **Explanation:** The lune is the difference between the larger circle's segment area and the smaller circle's segment area formed by their intersection. This formula requires knowing the distance $d$ between the centers of the circles, which depends on the specific configuration. **Summary:** The area of the lune bounded by two circles of radii $r$ and $R$ ($R > r$) intersecting at distance $d$ between centers is: $$\boxed{\text{Area} = \frac{R^2}{2}(\theta_R - \sin\theta_R) - \frac{r^2}{2}(\theta_r - \sin\theta_r)}$$ where $$\theta_R = 2\cos^{-1}\left(\frac{d^2 + R^2 - r^2}{2 d R}\right), \quad \theta_r = 2\cos^{-1}\left(\frac{d^2 + r^2 - R^2}{2 d r}\right)$$