# Circumference of Earth With Eratosthenes

• tchouhan
In summary, Eratosthenes used data to determine that the arc which separates two cities has a central angle of 7.2 degrees. He then used this information to calculate the circumference of the Earth. His method was to draw a circle and inscribe within two radii which are separated by an angle of 7.2 degrees.
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##

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

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

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).

SteamKing said:
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.

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.)

## What is the "Circumference of Earth With Eratosthenes"?

The "Circumference of Earth With Eratosthenes" is a method used to measure the circumference of the Earth by the ancient Greek mathematician and geographer, Eratosthenes. He used basic geometry and measurements to estimate the size of the Earth.

## How did Eratosthenes measure the circumference of the Earth?

Eratosthenes used the angle of the sun's rays at noon in two different locations to calculate the circumference of the Earth. He measured the angle of the sun's rays in his hometown of Alexandria, Egypt, and compared it to the angle in Syene, Egypt, which he knew was directly south of Alexandria and on the Tropic of Cancer. The difference in the angles allowed him to calculate the circumference of the Earth.

## How accurate was Eratosthenes' measurement of the Earth's circumference?

Eratosthenes' measurement was remarkably accurate for his time, estimating the circumference of the Earth to be within a few hundred kilometers of the actual value. However, with advancements in technology and more precise measurements, we now know that his calculation was slightly off, but still impressive considering the limited tools and resources available to him.

## What was the significance of Eratosthenes' measurement of the Earth's circumference?

Eratosthenes' measurement was significant because it was one of the first attempts to accurately determine the size of the Earth. It also provided evidence that the Earth was round, as it would not have been possible to measure the angles if the Earth was flat. His calculation also influenced future explorations and expeditions, as it gave a better understanding of the Earth's size and shape.

## How does Eratosthenes' method of measuring the Earth's circumference still impact us today?

Eratosthenes' method is still used as a basis for measuring the Earth's circumference and has influenced other methods of measuring the Earth's size and shape. His work also paved the way for advancements in geography, cartography, and other sciences. Additionally, his measurement of the Earth's circumference helped to shape our understanding of our planet and its place in the solar system.

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