In spherical geometry, the sum of the three interior angles of a spherical triangle is greater than 180 degrees by the spherical excess. This statement describes which relation?

• A. The sum equals 180 degrees
• B. The sum exceeds 180 degrees by the spherical excess
• C. The sum exceeds 360 degrees
• D. The sum equals 0 degrees

Answer: (B)

On a sphere, triangles feel the curvature of the surface, so their interior angles sum to more than 180 degrees. The extra amount beyond 180 degrees is called the spherical excess. The statement matches this relationship exactly: the total of the three interior angles equals 180 degrees plus the spherical excess. In formulas, α + β + γ = 180° + E, where E is the spherical excess. In radians, α + β + γ = π + E, and E equals the area of the triangle on the sphere divided by the square of the radius (for a unit sphere, E equals the area itself). This also explains why larger areas on the sphere produce larger excess, while tiny triangles in nearly flat regions have a very small excess.