Spherical excess in geodesy: For a spherical triangle on the celestial or terrestrial sphere, if S is the sum of its three interior angles, what is the spherical excess?

Introduction / Context:
On a sphere, the sum of the angles of a triangle exceeds 180°. The amount by which it exceeds 180° is the spherical excess, central to high-precision geodetic calculations and to understanding distortions on curved surfaces.

Given Data / Assumptions:

• S is the sum of the three internal angles (in degrees).
• The triangle lies on a spherical surface (Earth approximated as a sphere).
• Angles are measured in the same units throughout.

Concept / Approach:
For any spherical triangle, spherical excess E = S − 180°. This excess is proportional to the triangle’s area (on a sphere of radius R, Area = E * (π/180) * R^2). The larger the triangle, the greater the excess; for very small triangles, E tends toward zero, matching planar geometry.

Step-by-Step Solution:
Compute S by summing the three interior angles.Subtract 180°: E = S − 180°.Use E to check angular closure and to derive area if needed.

Verification / Alternative check:
In geodetic triangles spanning hundreds of kilometres, E may be several seconds to minutes of arc, consistent with Earth’s curvature.

Why Other Options Are Wrong:
Subtracting 90°, 270°, or 360° has no geometric basis for triangle sums; 180° − S would be negative for spherical triangles.

Common Pitfalls:
Forgetting unit consistency (degrees vs radians) when converting E for area; ignoring ellipsoidal corrections on precise networks.

Final Answer:
S − 180°.