# Sum of the angles of a spherical triangle

1. Jun 5, 2016

### harpf

1. The problem statement, all variables and given/known data
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.
2. Relevant equations
The text I have provides the following formula: sum of the angles = π + A/R^2

3. 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.

2. Jun 6, 2016

### Simon Bridge

I suspect not - though they are both dimensionless units, not all dimensionless units can be safely added ... consider degrees and radians.
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.
What is happening with the formula is that the number of radians you have to add to the plane geometry sum-of-angles happens to be the same as the number of steradians the area of the triangle subtends.

Try looking through a step-by-step:
https://nrich.maths.org/1434