Understanding Spherical Excess in Spherical Triangles

A spherical triangle is a triangle drawn on the surface of a sphere. Its sides are arcs of great circles of the sphere.

Unlike triangles in a flat plane, the sum of the interior angles of a spherical triangle is not fixed at $180^{\circ}$. Instead, the sum of the angles ($S$) of a spherical triangle is always greater than $180^{\circ}$ and less than $540^{\circ}$.

Defining Spherical Excess

The quantity obtained by subtracting $180^{\circ}$ from the sum of the angles ($S$) of a spherical triangle is a fundamental concept in spherical geometry.

Mathematically, this quantity is represented as:

$$ E = S - 180^{\circ} $$

This value, E, is known as the spherical excess.

Key Points about Spherical Excess

• The sum of the angles ($S$) is always greater than $180^{\circ}$, so the spherical excess ($E$) is always a positive value.
• The spherical excess is directly proportional to the area of the spherical triangle. The formula relating area ($A$) and excess ($E$) is $A = \frac{E}{360^{\circ}} \times \text{Area of sphere}$, or more commonly using radians, $A = R^2 E_{\text{radians}}$, where $R$ is the radius of the sphere and $E_{\text{radians}}$ is the excess in radians ($180^{\circ} = \pi$ radians).
• It quantifies how much the triangle's angle sum deviates from the planar triangle norm.

Why Other Options Are Incorrect

• Spherical error: This term is not standard terminology for the difference $S - 180^{\circ}$. Errors in spherical measurements are typically handled differently.
• Spherical deficiency: This term would imply the sum of angles is less than $180^{\circ}$, which occurs in hyperbolic geometry, not spherical geometry.
• Spherical misclosure: This term usually refers to the adjustment of errors in surveying, particularly in closing a traverse or loop, and is not directly related to the sum of angles of a single spherical triangle.

Therefore, the quantity $S - 180^{\circ}$ is specifically called the spherical excess.