London has coordinates (0.622515, 0, 0.782608) taking the earth as unit
radius.
Sydney has coordinates (-0.72509, 0.401925, -0.55919).
The straight line 'tunnel' distance between the two cities is
2d=1.943713
sinq =
dR
= 0.97186
Hence
q = 1.33300 radians = 76.38o.
The angle between the radii to the 2 places is 2.666 radians so, taking the
radius of the earth to be 6378 kilometres, the distance between London and Sydney
is 17,000 kilometres to the accuracy
of this calculation.
This distance is RQ where Q is the angle between the
radii to the 2 places. At an altitude of 6 kilometres the distance
is (R+6)Q and the 'extra' distance is 6Q = 16 kilometres.