# What are the angles of spherical triangles of a sphere partitioned in 4 equal spherical triangles?

A tetrahedron inside a sphere can divide a sphere into 4 equal spherical triangles. What are the angles, coordinates of vertices and arc lengths of those spherical triangles?

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

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