Two students who live on the equator want to estimate the circumference of the earth. The students synchronized their watches before one student leaves on vacation 600 miles due west of their home. Describe an experiment that these students can do to estimate the circumference of the earth.
Please see the attached diagrams, although note the scale is exaggerated for clarity. The two locations on the equator are labeled A and B. They are 600 miles apart. The center of the earth is labeled C. The black arrows indicate the direction of incoming sunlight. It is important to remember that the sun's rays are parallel to one another. This means that sunlight strikes the entire earth at the same angle at any given moment in time.
The students need to measure the angle of the sun in the sky in both locations at the same moment in time. The angle of the sun can be calculated using a vertical stick of known height (H) by measuring the length (L) of the shadow cast by the stick. Simple trigonometry is then used to compute the angle of the sun in the sky. Let's call the angle of the sun in the sky at the two locations XA and XB.
The difference of the two angles (XA-XB) is equal to the angle ACB. Now, to estimate the circumference of the earth, divide 360 degrees by angle ACB and then multiply by the distance between A and B (600 miles).
This is very similar to the procedure used by Eratosthenes - the first person to calculate the circumference of the Earth. Eratosthenes did not have the benefit of a friend helping him though, so he had to wait and do his calculation on a special day, the summer solstice, when he knew the angle of the sun in the sky would be zero at one location.