Hypothetical City's Foundation vs Curvature of the Earth

Question

If there is a City whose foundation is 1500 miles square, then how far down in our present earth's curvature would one have to excavate for its corners to remain ground level?

Thanks,

Kenneth Layne

Answer 1

Imagine the plane through the center of the Earth containing an edge of the city. On that plane you see a chord of length $2c = 1500$ miles on a circle of radius $r = 4000$ miles. Then the distance from the center of the Earth to the center of the chord is $ d = \sqrt{r^2 - c^2}$ so you would have to excavate $r-d$ miles at the center of each edge of the city.

At the center of the city you make the same calculation for a chord the length of the diagonal of the city: $1500\sqrt{2}$ miles.

Google says

4000 - sqrt(4000^2 - 2*750^2) = 143 miles

at the center of the city.