Help with Eratosthenes calculation?

Question

Distance from Aswan to Alexandria about 500 miles
sun at noon from each town - finding the angular difference to be 7 degrees 12 mins. How do I calculate this?

the answer is 25,000 miles

Answer 1

I found this webpage to be most helpful:

http://www.youth.net/eratosthenes/

The formula is:

D/d = A/a

where D is the diameter of the earth, d is the distance between Aswan and Alexandria, A is 360°, and a is the angle you supplied.

a = 7° 12' = 7 12/60° = 7 1/5° = 7.2°

D = dA/a = 500 miles (360°/7.2°) = 500 miles (50) = 25,000 miles.

Answer 2

You know that the Earth is almost a circle. Therefore, the full complete Earth has 360 degrees.

The distance from Aswan to Alexandria is about 500 miles and is 7 1/5 degrees apart. (You can convert the 12 minutes into a fraction by dividing by 60).Therefore, you know that 7 1/5 degrees of the circle (earth) is about 500 miles.

Now, you figure out that one degree of Earth is 500 / (7 1/5) which is 69 4/9. Now you can multiply it by 360 to get the circumference of the full circle, which comes out to be 25000 miles, which is pretty close to the modern estimate of 24,901.55 miles.

That's how Eratosthenes, an ancient Greek mathematician, figured out the approximate circumference of the Earth over 2,000 years ago.

Answer 3

If I understand Eratosthenes' calculation correctly, he surmised that the angular difference in the zenith of the sun at noon in the two towns (assuming that the towns were directly north-south) gives you the circumference of the earth.

Using the numbers in your problem, two towns lined up north-south have the sun at 7 degrees 12 minutes (= 7 and 12/60 degrees, or 7.2 degrees) difference. Because there are 360 degrees in the circumference of the earth running through those two towns, this indicates that the towns are 7.2/360 = .02 of the earth's circumference apart.

That distance is 500 miles per your problem, so
.02*c = 500
c = 500/.02 = 25000 miles as the circumference of the earth.

Hope that helps.

Answer 4

7degrees 12 minutes = 7.2 degrees = 500 miles
So (360/7.2)*7.2 = (360/7.2)* 500
360 degrees = 50*500 = 25,000 miles. This was his estimate for the circumference of the Earth.

His reasoning was that the Sun's rays were parallel so that the angular difference was the same as the central angle formed by lines from the center of the earth to each city. He also assumed that the two cities were on the same latitude.

Answer 5

I'll answer it sort of, from the point of view of a US Citizen. I live on the East Coast. I know the West Coast is 3000 miles away approximately. I also know that Pacific Standard Time is three hours different than Eastern Standard Time. This makes each time zone represent about 1000 miles.

Since each day has 24 hours, there are 24 different time zones around the earth. Therefore the cirucumference of the earth is 24 time zones times 1000 miles per time zone = 24,000 miles