Circumference of Earth With Eratosthenes

[tchouhan]

Homework Statement

Eratosthenes measured the circumference of the Earth by noting that the Sun is at an angle of 7°12' = 7.2° ("one-fiftieth of a circle") south of the vertical in Alexandria at the same time of day and year that it is directly overhead in Syene. Syene is 5000 stades directly south of Alexandria. (The stade was a Mediterranean unit of length that varied slightly from region to region, but in Egypt was most likely equal to 157.5 meters.) Find the circumference of the Earth from these data. The Earth's equatorial circumference is 40075 km according to NASA. What was Eratosthenes percent error?

Homework Equations

$a^2+b^2=c^2$

Conversion between stades/meters.

sin, tan, cosine perhaps?

The Attempt at a Solution

So I've gone ahead and tried to draw a diagram, and I tried to do the $tan(7.2)=x/787500$ but the number I got was incorrect. I got 787,500 meters by multiplying (5000)(157.5), since 1 stade is 157.5 meters, so I figured out that 5000 is 787,500 meters or 787KM

 

[SteamKing]

What part of a circle is an arc of 7.2 degrees? Instead of degrees, think radians.

 

[rude man]

I'm not going to add to this except to point out that the Ancients knew the Earth was round LONG before Chrissie C. sailed the ocean-blue. (It really was blue then, now it's a continuum of pollution).

 

[tchouhan]

What part of a circle is an arc of 7.2 degrees? Instead of degrees, think radians.

so $arctan(7.2)$? I am confused as to what you mean by what part of a circle is 7.2 degrees.

 

[SteamKing]

It's very simple. Syene lies due south of Alexandria a distance of 5000 stadia. By measurement of angles at these two locations, Eratosthenes determined that the arc which separates the two cities has a central angle of 7.2 degrees. Eratosthenes wants to determine the circumference of the Earth using these data.

To clarify his method, draw a circle and inscribe within two radii which are separated by an angle of 7.2 degrees.

(Hint: You are trying to determine the circumference of a circle, not figure out the sides of a triangle.)