Task
The ancient Greek scientist Eratosthenes devised the following experiment for estimating the circumference of the earth, which he assumed to be spherical in shape. Pictured below is the sun's rays hitting two different locations on the surface of the earth. The point $P$, in the picture below, is in the tropics and, at this particular time, the sun's rays are hitting this point perpendicularly. At the point $Q$, the sun's rays meet the earth at an angle which can be measured by finding the length of the shadow cast by an object at point $Q$.

Assuming the rays of the sun hitting points $P$ and $R$ are parallel as in the picture, explain why angle $POQ$ is congruent to angle $QRS$.

For the two locations used by Eratosthenes, the shadow cast by a ten foot pole at location $Q$ was about 1.26 feet. Using this information, find $a$, the measure of angle $POQ$.

According to Eratosthenes, the distance from point $P$ to point $Q$ was
approximately $2,584,000$ feet. Using this information and the calculation
from part (b) what estimate does this give for the circumference of the earth, in both feet and miles?

Current estimates for the earth's circumference are about 24,900 miles.
Within what percent error from this current value is Eratosthenes' estimate?
IM Commentary
The picture has been drawn with $a = 20$ because the $7.2$ degree angle
which would accurately represent the measurements taken by Eratosthenes
was too small to allow space for all of the designated points. Also of course the points $R$ and $S$ are not intended to be ''realistic'' as in practice the height
of the object casting a shadow will be very small compared to the circumference of the earth.
The
accuracy and simplicity of this experiment are amazing. A wonderful project
for students, which would necessarily involve team work with a different school
and most likely a school in a different state or region of the country, would be
to try to repeat Eratosthenes' experiment. Since the Continental United States does not
have any land within the tropics, students would either have to collaborate with
a school in Hawaii or alter Eratosthenes' method. What happens to this method
if point $P$ is moved to the other side of $Q$?
Teachers may wish to discuss some of the hypotheses behind this method. For example, is it reasonable to assume that the rays of sun reaching points $P$ and
$Q$ are parallel? How do you measure the distance from $P$ to $Q$? With GPS
systems this is less of an issue today but for Eratosthenes it must have presented a formidable challenge. Teachers may also wish to give more of a historical
context to this problem. The two cities where measurements were taken were
Syene (modern Aswan in southern Egypt) and Alexandria (also in Egypt)
at noon of the summer
solstice: this way the sun was directly overhead at this moment at one of the two locations, namely Syene. Since Alexandria is directly north of Syene, Eratosthenes was estimating
the polar circumference of the earth which is slightly smaller than the equatorial
circumference.
This task is intended mainly for instructional purposes, giving an interesting
context for implementing ideas from geometry and trigonometry.