Eratosthenes knew that in Aswan the sun was directly overhead at noon on the summer solstice.
At the same time, he measured the angle of the sun to a vertical line in Alexandria. He found this
angle to be one(1/50) fiftieth
of a circle
a. What was Eratosthenes's measurement in degrees? (Hint: There are 360cin a complete
circle.)
b. What can you conclude about the angle formed between the two cities and the centre of the
Earth?
c. Suppose Eratosthenes estimated the distance from Aswan and Alexandria to be 900 km.
What circumference would he have found for the Earth?
a. To find Eratosthenes's measurement in degrees, we need to find the angle made by 1/50 of a circle. We know that a complete circle is 360 degrees, so we can find the measurement using the following calculation:
Angle in degrees = (1/50) * 360
Angle in degrees = 7.2
Therefore, Eratosthenes's measurement in degrees is 7.2 degrees.
b. From Eratosthenes's measurements, we can conclude that the angle formed between the two cities (Aswan and Alexandria) and the center of the Earth is 7.2 degrees. This angle represents the difference in the direction of the sun's rays between the two cities at noon on the summer solstice.
c. Suppose Eratosthenes estimated the distance from Aswan and Alexandria to be 900 km. To determine the circumference of the Earth, we can use the following formula:
Circumference of Earth = (Distance between Aswan and Alexandria) / (angle in degrees * 360)
Circumference of Earth = 900 km / (7.2 degrees * 360)
Circumference of Earth ≈ 39,687.5 km
Therefore, using these measurements, Eratosthenes would have estimated the circumference of the Earth to be approximately 39,687.5 km.