Sum of the angles of a spherical triangle

In summary, the sum of the angles of a spherical triangle formed on a sphere with radius R is equal to π plus the area of the surface of the sphere enclosed by the triangle divided by R squared. This is derived from the formula for the sum of angles on the unit sphere, where the addition of "unit sphere" adds the implied scale factor of R=1<unit>. This is also how radians are defined, as the arc length on the unit circle inside the angle. The number of radians that need to be added to the plane geometry sum-of-angles is the same as the number of steradians that the area of the triangle subtends. Therefore, the right side of the formula is not a sum of radians and ster
  • #1
harpf
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0

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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  • #2
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
 

1. What is the sum of the angles of a spherical triangle?

The sum of the angles of a spherical triangle is always greater than 180 degrees and less than or equal to 540 degrees. This is because the surface of a sphere is curved, so the angles do not follow the same rules as a flat surface.

2. How can I calculate the sum of the angles of a spherical triangle?

To calculate the sum of the angles of a spherical triangle, you can use the following formula: Sum = 180 degrees x (n - 2), where n is the number of sides or angles in the triangle. For example, if the spherical triangle has 4 sides, the sum of the angles would be 180 degrees x (4-2) = 360 degrees.

3. Is the sum of the angles of a spherical triangle affected by the size of the triangle?

No, the sum of the angles of a spherical triangle is not affected by the size of the triangle. This is because the angles are measured in relation to the curvature of the sphere, not the actual size of the triangle.

4. Do all types of spherical triangles have the same sum of angles?

Yes, all types of spherical triangles have the same sum of angles. This is because the sum is determined by the number of sides or angles in the triangle, not the type of triangle it is.

5. Why is it important to understand the sum of the angles of a spherical triangle?

Understanding the sum of the angles of a spherical triangle is important for various fields of study, such as geography, astronomy, and navigation. It helps us understand and calculate distances and angles on a curved surface, which is essential for accurate measurements and calculations in these fields.

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