1. **Problem Statement:** We need to state and prove the relationship between the internal angles of a spherical triangle and the area of that triangle. 2. **Background:** A spherical triangle is formed by three great circle arcs on the surface of a sphere. Unlike planar triangles, the sum of the internal angles of a spherical triangle exceeds $180^\circ$ (or $\pi$ radians). 3. **Key Formula:** The spherical excess $E$ is defined as: $$E = (A + B + C) - \pi$$ where $A$, $B$, and $C$ are the internal angles of the spherical triangle in radians. 4. **Relationship to Area:** The area $S$ of a spherical triangle on a sphere of radius $R$ is given by: $$S = E \times R^2 = ((A + B + C) - \pi) R^2$$ This means the area is proportional to the spherical excess. 5. **Proof Sketch:** - Consider the sphere of radius $R$. - Each angle corresponds to a sector of the sphere. - The sum of the angles exceeds $\pi$ by the spherical excess $E$. - The area of the spherical triangle is exactly $E R^2$. 6. **Explanation:** This formula shows that the larger the sum of the internal angles beyond $\pi$, the larger the area of the spherical triangle. This is a fundamental result in spherical geometry, contrasting with planar triangles where the sum of angles is always $\pi$ and area depends on base and height. **Final answer:** $$\boxed{S = ((A + B + C) - \pi) R^2}$$ where $S$ is the area, $A$, $B$, $C$ are the internal angles in radians, and $R$ is the radius of the sphere.