How have we measured the radius of the Earth (old and new)? I'd imagine ancient methods are fun exercises in geometry. how has the measurement has evolved over time?

The first method is credited to Eratosthenes. He knew that during the summer solstice in Syenne the angle of the Sun overhead at noon was $0^\circ$. Following this he made the same measurement in Alexandria and measured the angle of the Sun overhead using shadows and calculated it to be $\approx 7^\circ$ which is approximately $1/50$ of a circle. Hence, both the circumference and the radius are easily calculated and quite accurate for such crude methods.
This is a pretty broad question so I'll limit myself to describing one instance of how the radius of the Earth is measured. Specifically, I'll talk about how it's measured today, with current technology.

As technology has progressed, our ability to precisely measure facets of the Earth has drastically improved. At this point, we can measure the exact shape and size of the Earth to such precision that asking about the "radius of the Earth" is no longer meaningful. Primarily because we know the shape of the Earth to such precision that we know it is not a perfect sphere with a fixed radius.

The current standard used to describe the exact shape of the Earth is the World Geodetic System, the latest edition of which is referred to as WGS84. In effect, this is a coordinate system centered on the Earth that precisely describes the shape of the Earth. This is done by using sensitive satellite measurements to fit the spherical harmonics to the Earth's geoid. The current, most updated version of WGS84 has more than 4.6 million harmonic terms with an accuracy in "radius" at any given point of 10 km. That is, you could pick any precise location on the Earth and use WGS84 to calculate the radius of the Earth at that exact point to an accuracy of 10 km. This radius would of course vary as you moved about the Earth.

If you really want to break this down into a simple number, the WGS model, at the crudest level, considers the Earth to be an oblate spheroid with an equitorial radius of $R_{eq} = 6\:378\:137\:\mathrm{m}$ and a polar radius of $R_{p} = 6\:356\:752.3\:\mathrm{m}$.