Apologies if this question has been asked before.
Across the surface of the Earth, the distance between London and New York is 5567 km. Given that the earth has a radius of 6371 km, what is the distance between London and New York, supposing a path between them was dug through the earth's crust?
I am interested because I would like to know the minimum theoretical latency for data transmission between the two cities (given the speed of light), but I don't know how to calculate it myself. (I made an attempt and got 9770km, which is obviously wrong…)
Best Answer
This means $\theta$, your angle between lines extending from Earth's centre to London and New York, is equal to $\frac{5567}{6371}=0.8738$ radians.
Now let's construct a triangle with your proposed tunnel, the radius from Earth's centre to NY, and from Earth's centre to London. The angles in a triangle add to $\pi$ radians, and since two of the legs of the triangle (the two radii) are the same, we know the two remaining angles are the same. Thus, $\frac{\pi-0.8738}{2}=1.1339$ is the measure of the other two angles.
Using law of sines and letting $x$ be the length between NY and London:
$$\frac{\sin1.1339}{6371}=\frac{\sin0.8738}{x}$$
$$x=\frac{6371\sin0.8738}{\sin1.1339}=5391.5475$$
So the distance between New York and London is $5391.5475$ km.