# Sum of the angles of a spherical triangle

## Homework Statement

What is the sum of the angles of a spherical triangle formed on the surface of a sphere of radius R? The triangle is formed by the intersections of the arcs of great circles. Let
A be the area of the surface of the sphere enclosed by the triangle.

This question is a result of self-study.

## Homework Equations

The text I have provides the following formula: sum of the angles = π + A/R^2

## The Attempt at a Solution

A course I had last year covered steradians. My confusion relates to the formula. If the triangle was two-dimensional, the sum of the angles would of course be π radians. Also, the surface area enclosed by the spherical triangle subtends a solid angle of A/R^2 steradians.
Do these details mean that the right side of the formula listed above is a sum of radians and steradians? Are radians and steradians both just considered “degrees” that can be added together?

Thank you for clarifying.

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Simon Bridge
You should go through the derivation carefully - for instance, the formula for the sum of angles on the unit sphere would be $S=\pi+A$ ... clearly you cannot add an angle and an area: the dimensions don't match. What has happened is that the addition of "unit sphere" in the description adds the implied scale factor that R=1<unit>. This is also how radians get defined: the arcength on the unit circle inside the angle.