Measuring the circumference of the earth - revisited I have done a bit of research on this topic already and the easiest way seems to be the experiment by Eratosthenes. We pick two locations on the same latitude (of which we know the distance) and then measure the length of the shadow of a vertical stick at solar noon to get two angles. We can then figure out the difference between these angles and plug it into the formula:
\begin{align}
\frac{\text{angle}}{360°}&amp;=\frac{\text{distance}}{\text{circumference of the earth}}\\[3mm]
\text{circumference of the earth}&amp;=\frac{360°}{\text{angle}}\times\text{distance}\\
\end{align}
Now I would like to do a similar experiment. My family has friends who live 1260km west (and a little south) from us. Is there a way for us take a measurement at the same time and calculate the circumference of the earth? Obviously the formula above does no longer apply since we never have the same solar noon.
You can use the fact that the Earth's rotation period is known.  Then divide that period (use the solar day of 24 hrs 00 minutes instead of the sidereal day of 23 hrs 56 minutes) by the actual amount of time between your two solar noons and multiply the resulting factor by your 1260 km distance.
The result is the circumference ... of your latitude circle.  To get the circumference of a great circle and thus the Earth, you need to divide the calculated number above by the cosine of your latitude.  To get the latitude perform your experiment on an equinox and measure the shadow angle at solar noon.
Have fun, good luck!