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A tetrahedron inside a sphere can divide a sphere into 4 equal spherical triangles.

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What are the angles, coordinates of vertices and arc lengths of those spherical triangles?

Bear in mind link:
Since the sides of a spherical triangle are arcs, they can be described as angles, and so we have two kinds of angles:

  1. The angles at the vertices of the triangle, formed by the great circles intersecting at the vertices and denoted by Greek letters.
  2. The sides of the triangle, measured by the angle formed by the lines connecting the vertices to the center of the sphere and denoted by lower-case Roman letters.

enter image description here

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  • $\begingroup$ side $\arccos(-\frac13)$ and angle $120^\circ$. $\endgroup$
    – achille hui
    Apr 3, 2018 at 12:16
  • $\begingroup$ Can you specify which angle you refer to using the notation of the image above? $\endgroup$
    – RutgerH
    Apr 3, 2018 at 12:26
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    $\begingroup$ $a = b = c = \arccos(-\frac13)$ (assume the sphere is a unit sphere) and $\alpha = \beta = \frac{2\pi}{3}$. $\endgroup$
    – achille hui
    Apr 3, 2018 at 12:27
  • $\begingroup$ I think you can find the coordinates of the vertices simply by taking the coordinates of the vertices of a regular tetrahedron. $\endgroup$
    – M. Winter
    Apr 3, 2018 at 12:29
  • $\begingroup$ See this question ... for something similar ? math.stackexchange.com/questions/859978/… $\endgroup$
    – Donald Splutterwit
    Apr 3, 2018 at 12:43

1 Answer 1

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Answering your title question, the angles are, by symmetry, $2\pi/3$.

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  • $\begingroup$ are those the angles denoted by the Greek characters? Can you specify which angle you refer to using the notation of the image above? $\endgroup$
    – RutgerH
    Apr 3, 2018 at 12:25

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