Astronomy Measuring the Diameter of the Earth

Astronomy Measuring the Diameter of the Earth Eratosthenes' Method Eratosthenes made a calculation of the Earth's diameter, based on one assumption and two measurements. The assumption was that the Earth was a sphere. This assumption is required since the method relies on accepting that the Earth's diameter is a total of 360o. The two measurements made are the degree of the Earth's shadow at noon at two points and the distance between those points. Eratosthanes used the city of Syene in Egypt as the first point.

This point was selected because it was known that on noon on the first day of summer the sun was directly overhead. This was known because people observed that at this time, the buildings cast no shadows (York University). Therefore, the degree of the shadow at Syene was 0o. Eratosthanes then needed to know the degree of the shadow at another point either directly north or directly south, at the same time of day. Eratosthanes selected Alexandria as the second city.

The degree of the sun's shadow was measured and found to be 7.2o (HEASARC). Eratosthanes then knew that the degree of difference between Syene and Alexandria was 7.2o. The final item Eratosthanes needed to know was the distance between the two cities. This was done by sending a runner to pace out the distance (York University). Eratosthanes found that the distance between the two cities was 5000 stadia. A stadium was an ancient measurement, thought to equal around 0.15 kilometers (HEASARC).

From these measurements, it is known that 7.2o is equal to a distance of 5000 stadia. Since the circumference of the Earth is 360o, the distance will be equal to: 360/7.2 x 5000 = 250,000 stadia. Since a stadia is thought to be 0.15 kilometers, the circumference of the Earth is equal to: 250,000 stadia x 0.15 km = 37,500 kilometers.

The radius of the Earth can then be calculated by using the formula for the circumference of a circle, C = 2?r: 37,500 km = 2 x 3.14 x r Since the diameter of a circle, is twice the radius, the diameter will be equal to: 5971 km x 2 = 11942 km. Therefore, Eratosthanes calculated the diameter of the Earth to be 11942 kilometers. One important point in the experiment is that one of the locations does not need to have a shadow of 0o for the experiment to be successful.

The calculation requires that the difference in the angle of the shadow at two locations be known. Eratosthanes simply made the experiment a little simpler by having the first location as a 0o shadow. This meant that only one measurement of shadow had to be made. How to Measure the Diameter of the Earth This experiment can be repeated by measuring the degree of the shadows cast at two locations either directly north or south of each other at noon on the same day.

Before beginning the experiment it is necessary to organize with someone at the other location to take the measurement at noon. This measurement will be required to calculate the diameter of the earth. The other location, now to be referred to as location B, will complete the exact same measurements as your location, now to be referred to as location A. The completion of these measurements will now be described. Equipment Required meter ruler. Measuring tape. Calculator. Watch. Atlas or directory to determine the distance between points A and B.

Measurements Note: actual measurement must be completed at noon, allow adequate time to setup for the measurement. 1. Prior to noon, take a 1 meter ruler and hold it perpendicular to the Earth's surface. Ensure that the angle between the ruler and the ground is 90o. 2. Mark the point where the meter ruler touches the ground. 3. At noon, observe the shadow that the meter ruler makes. Mark the point where the shadow touches the ground. 4.

Measure the distance between the point where the ruler touches the ground and the point where the shadow touches the ground. Record the distance in cm. 5. Since the ruler is 1 meter high, you now know the distance of two lengths of a right-angle triangle. The diagram below depicts the experiment and measurements. Diagram 1: Calculating the angle of the shadow.

Use this information to calculate the angle of the shadow as follows: Tan ? = O / A Therefore, ? = O / A Tan-1 Where, O = distance as recorded in step 4 and A = 100 cm (length of ruler). The calculated value is the degree of.