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.

source: Self made

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.

Image source:A Survey of the Solar System

Let radius of Earth be R.
Now circumference of a circle(Earth in this case) is given by 2piR

• i.e. 360 deg subtended at the center by the arc means length of arc= 2piR
• 1 deg subtended at the center by the arc means length of arc= 2piR/360
• 7.2 deg subtended at the center by the arc means length of arc= 2piR*7.2/360 but the length of the arc is equal to the distance between Alexandria and Syene!

So we get:

The distance between Alexandria and Syene was 787km and hence he calculated the radius R to be about 6,267 kilometers and the circumference of Earth to be 39,350km. The actual radius of the earth is 6,378.1 kilometers at the equator, but the radius varies slightly because the earth is not a perfect sphere but flattened at the poles. The radius is about 6,371 kilometers at the poles. He was off just by about 2 percent 2000 years ago!
Note that, you don't one of the places to have perpendicular rays. You can just subtract whatever angle the sunlight is between the two points and replace 7.2 by A=(l-m) as shown below!


image source

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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.

image source:owlcation

Then the height can be determined as:


Once he had the height, he went to the top of the mountain and measured the angle of dip of the Horizon.


image source:By Nevit Dilmen (talk) - Own work Using Geogebra and Inkscape, CC BY-SA 3.0, Link

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:

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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:

• Stand and Facethe eastern horizon and wait for the tip of the sun to be visible.
• As soon as you see it, start your stopwatch and lie down. Lying down will make the sun disappear for a few seconds.
• Now stop the stopwatch when you see the top limb again.

For sunset the idea is again similar.

• lie down flat facing the sun and wait for it to set.
• Just when you see the tip of the sun disappear, start the stopwatch and stand up. Standing up will make the sun visible again.
• Now wait for the sun to disappear again, and as soon as it does, stop your stopwatch.

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.

image source

However because the Earth is rotating, after some time you will again be able to see the sun!


image source

In the time interval between you starting and stopping the stopwatch, the Earth rotates by an angle theta!
Therefore, knowing the time will tell you the theta that Earth rotated.

• In 1 day or 86400sec, Earth rotates 360 deg in
• In 1 sec, Earth rotates in 360/86400 deg
• In t sec, Earth rotates theta = t x 360/86400 deg
  
  
  
  Knowing your height H, the Radius R can be easily found from basic trignometry as
  
  

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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References

• How to Find the Radius of the Earth
• Measure the Earth's Radius with a Stopwatch
• History of geodesy

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