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Sum of the angles of a spherical triangle

  1. Jun 5, 2016 #1
    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. jcsd
  3. Jun 6, 2016 #2

    Simon Bridge

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    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
     
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