Final answer:
Girard's Theorem states the area of a spherical triangle is r²(E - π), where E is the sum of the spherical triangle's interior angles, and Pi represents the mathematical constant π.
Step-by-step explanation:
Girard's Theorem states that the area of a spherical triangle on a sphere of radius r with interior angles a, b, and c is given by the formula:
Area = r²(E - π)
where E is the sum of the interior angles of the triangle (a + b + c), and π is the mathematical constant Pi. To understand this further, if we consider the interior angles of a spherical triangle that exceed the sum of angles in a flat, Euclidean triangle by an amount known as the spherical excess, which is E - π. For example, if we have a spherical triangle with interior angles a = 60°, b = 60°, and c = 60°, the spherical excess would be 60° + 60° + 60° - 180° = 180° - 180° = 0°, indicating that such a triangle would not have an area on a sphere with radius r.
Applying this theorem allows us to calculate the area of a spherical triangle accurately, accounting for the curvature of the sphere.