Measuring Earth's Radius from Angle to Horizon

Here, we will determine how to measure the radius of the Earth by measuring the angle to the horizon. Assume we are at the beach and can see a clear horizon line where the sky meets the water. We will call the radius of the Earth r. We stand a height h above sea level. We measure a distance θ to the horizon. Since the tangent line is perpendicular to the radius of the Earth, we can deduce the following geometry:

We can see that θ is equal to the angle between the two radius vectors. Taking the cosine of this angle, we can deduce the following:

Since the angle down to the horizon will be very small, this 2nd order Taylor approximation for θ is valid. 

Knowing that the true radius of the earth is between 10^6 and 10^7 meters, we can find h as a function of θ.  This will tell us how high we must elevate ourselves, given the minimum angle we are capable of measuring. Solving for h,

Using this equation, we can construct a table for the value of h needed, corresponding to the minimum θ capable of being measured, for r = 10^6 m and for r = 10^7 m:

Assuming that the radius of the earth is 10^7 m (worst case scenario), and that we are capable of measuring 0.5° of angular displacement to the horizon, we see that we must achieve an elevation of 380 m to accurately perform the measurement.