Earth is so large, it is hard to see any curvature of the Earth whatsoever, even when standing on a beach. Then, how do you measure the radius of the Earth?(Assuming you are not one of the Flat-Earthers!)
The oldest documented method to measure the radius of Earth was given by Erasthosthenes in the Third Century BC . Let that sink in. Third century BC!
Erasthosthenes was in the city of Alexandria in Egypt then. He came to know of a unique well in Syene. On a particular day at noon, one could see the reflection of the sun at the bottom of the deep well. This meant that the sun was directly overhead or the rays of light were exactly perpendicular to the surface of the Earth at that point. At the same time, however, there was a shadow cast by any tall object in Alexandria from which he could find the angle at which light from the sun was coming as shown below.
So the angle is given by, arctan(L/H). This angle is same as the angle subtended at the center of the Earth by these two places. He found that the angle was about 7.2degrees.
Let radius of Earth be R.
Now circumference of a circle(Earth in this case) is given by 2piR
So we get:
Al Biruni, improved the method of measurement of radius in 10th century AD. Al Biruni used an Astrolabe to measure the angle of elevation and angle of dip to measure the radius.
His method's first step was to measure the height of a mountain. To do so he measured the angle of elevation of mountain from two places and measured the distance between the two places.
Now from the angle of dip and by using the law of sines, Al Biruni found the radius of Earth with just about 1percent error.
Solving we get:
Michael Richmond discussed this easy method to approximate the radius of Earth. All you need for it is a stopwatch and yourself!
If you are on a beach facing East, you can do this experiment during sunrise, and if you have a beach facing west you can do this during sunset.
For sunrise:
For sunset the idea is again similar.
The sun rising and setting is all because of the rotation of earth. The Earth completes one rotation in about 24 hrs i.e. it rotates by 360 degrees in 24hrs. Night is when the light from the sun doesn't reach you which is because of the shadow of the earth! Standing lets you peek above the shadow as shown below. If you lie down then the sun gets hidden.
Apart from these there have been several modern methods of measuring the radius of earth. You can find some of them here.
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