Question 1184188: Please help me to solve this
Eratosthenes of Cyrene was a Greek scholar whole lived and work in Cyrene in Alexandria. One day while visiting Syene, he noticed the sun ray shone directly down a well. On this date, 1 year later in Alexandria which is 500 miles due north of Syene, he measured the angle of the sun about 7.2 degrees using this information, approximate the radius and the circumference of the Earth.
Answer by ikleyn(52787)   (Show Source):
You can put this solution on YOUR website! .
This story and this problem are well known.
The story came to the forum long time ago, and the problem was solved here under this link
https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1165118.html
So, I simply copied and paste it here AGAIN for your convenience.
By the way, in that old post, the problem was worded / presented much better than here in yours.
So, I brought this old formulation, too . . .
E N J O Y ( ! )
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circumference of the Earth.
The Greek mathematician Eratosthenes measured the circumference of the Earth from the following observations.
He noticed that on the certain day the sun shone directly down a deep well in Syene.
At the same time in Alexandria, 500 miles north, the rays of the sun shone at an angle of 7.2 degrees to the zenith.
Use this information and the figure to find the radius and circumference of the Earth.
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The Greek mathematician Eratosthenes was smart enough to INTERPRET these observations in a right way.
In Syene, the Eartn radius at that day and at that time was directed EXACTLY to the Sun.
In Alexandria, 500 miles North, the Earth radius (at the same day and the same time) made the angle of 7.2 degrees
with the direction to the Sun.
It means that the directions of the two radii made the angle of 7.2 degrees.
Then Eratosthenes used this equation for the arc of a circle
S =  .
Here S is the arc length, R is the Earth radius (which should be evaluated/estimated), and  is the angle between two radii
drawn to the arc's endpoints (central angle) in radians.
(!) ancient Greeks and Egyptians just knew it at that time (!) (!)
In this equation, S = 500 miles and = 7.2 degrees =  =  = 0.1256 radians.
Thus R =  =  = 3981 miles (rounded).
The contemporary value is R = 3959 miles
(see this Wikipedia article
https://www.google.com/search?q=the+eart+radius&rlz=1C1CHBF_enUS910US910&oq=the+eart+radius&aqs=chrome..69i57j0l7.6786j0j15&sourceid=chrome&ie=UTF-8 )
So, the Eratosthenes' estimate is very good (fantastically good, I'd say).
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About Eratosthenes and his great achievements, read these Wikipedia article
https://en.wikipedia.org/wiki/Eratosthenes
or see and listen this video
https://www.businessinsider.com/how-greek-eratosthenes-calculated-earth-circumference-2016-6
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