Each triangular face of the tetrahedron is projected onto the surface of the sphere so that each of the angles of the spherical triangle is 120°.

 cos C= - cos A cos B + sin A sin B cos c 

 

so cos c = (cos C + cos A cos B) / (sin A sin B)

cos c = (cos 120 + cos^2 120) / sin^2 120

(-1/2 + 1/4) / (3/4) = (-1/4)*(4/3) = -1/3

indicating the cosine of a side of the spherical triangle is -1/3.

That is the cosine of the central angle for the edge of the triangle containing the side required.  Then, by the plane law of cosines

s^2 = 1 + 1 - 2*(-1/3) = 2 + 2/3 = 8/3

s = sqrt(8/3) =~ 1.63299316185545