Expressing latitude and longitude in radians:

Athens: *P*_{1} = 38° N = 0.6632251;

*Q*_{1} = 24° E = 0.418879.

Hong Kong: *P*_{2} = 22° N = 0.3839724;

*Q*_{2} = 114° E = 1.989675

We now find the *x*-, *y*-, and *z*-coordinates for Athens and Hong Kong, given that the radius of the Earth is 6371 km:

*x*_{1} = 6371 cos 0.6632 cos 0.4189 = 4586.4

*y*_{1} = 6371 cos 0.6632 sin 0.4189 = 2042.1

*z*_{1} = 6371 sin 0.6632 = 3922.3

*x*_{2} = 6371 cos 0.3840 cos 1.9897 = -2402.7

*y*_{2} = 6371 cos 0.3840 sin 1.9897 = 5396.3

*z*_{2} = 6371 sin 0.3840 = 2386.8

Now for the straight line distance between the 2 cities (directly, through the Earth), using the 3-D distance formula:

`sqrt ((-2402.7- 4586.4)^2+ (5396.3-2042.1)^2+ (2386.8-3922.3)^2)` `= 7902.9" km"`

Next, we find the central angle.

`7902.9/2=3851.5`,

and since `sin theta = 3851.5/6371` and the central angle is twice *θ*, we have:

Central angle = `2arcsin(3851.5/6371)` `= 2 xx 0.64918 ` `= 1.29836`

(Radians, of course, and full calculator accuracy was used throughout, but not shown.)

We use

s=rθ

giving

s= 6371 × 1.29836 = 8272 km.

[Thanks to reader Paul Holland for the above example.]

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