How do we measure the curvature of space due to earths gravity since we can't dig a hole through the centre of the earth and measure the actual diameter of earth which is slightly different (1.8cm) from the ratio of circumference and pi. I think I have read somewhere that Gauss was the first one to measure this curvature. Furthermore the earth isn't that much of a perfect sphere to determine the diameter with too much accuracy by dividing the circumference by pi. So how can we figure out the curvature by looking at the difference between actual diameter and (circumference/pi) if an accurate diameter of earth does not exist due to: 1.Earth's slightly elliptical shape. 2. Mountains of different sizes on the surface of earth.
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Satellites, laser ranging, atomic clocks, very long baseline interferometry, and improved models of the Earth's gravitational field.
A couple of decades ago, improved measurements of the solar system allowed (and forced) developers of planetary ephemerides to incorporate general relativity into their models. (It would have been silly to do so prior to that. The measurement errors were orders of magnitude larger than the corrections offered by using general relativity.)
The same thing is happening now in geodesy. Ever improving measurements and models are pushing the boundaries of what can be done using Newtonian assumptions. The measurement errors will soon be below the errors that result from those Newtonian assumptions. Improvements will demand abandoning those Newtonian assumptions. Here's a recent paper that suggests doing exactly that:
answered Jan 20, 2015 at 21:16