Finding the Earth's Circumference

This calculation requires measuring the angle between a vertical post or stick (called a "gnomon") and the Sun's rays at the time when the Sun is at its highest point in the sky (called "local noon"). This angle (called the Sun angle) is then compared to the same measurement made by another lab team at a place located geographically north of your own latitude and as close to your own longitude as possible. The two experimental measurements should be made within one calendar day of each other, but they do not have to be made on the vernal (or autumnal) equinox. A relatively simply mathematical ratio is then used to calculate the Earth's circumference.

Since the measurement of the Sun's angle at local noon is the key to this collaboration, several methods are suggested depending on the mathematical ability of your students:

1. For students who understand simple right triangle trigonometry
Place a vertical stick or post (the gnomon) of known height in the ground making sure both that the stick or post is perpendicular, and that the surface it is placed on is relatively flat. Beginning at least twenty minutes before noon (12 pm) mark the maximum position of the shadow cast by the gnomon every two or three minutes until at least twenty minutes after noon. The time the shadow was the shortest is local noon. Now just divide the shadow length by the height of the gnomon to find the tangent of the sun angle. Now use the arctangent to find the sun angle itself.

2. For less mathematically sophisticated students
A less sophisticated method uses a smaller stick or post placed approximately 50 cm north of the gnomon, such that the shadow cast by the taller stick falls on the smaller stick. Local noon is the exact time of this alignment. Determination of the angle of declination is then made directly by stretching a string from the top of the gnomon to the tip of the shadow and reading the angle from a protractor placed at that point. This method, while lacking some precision, is a good approximation and provides the student with a more clear picture of the Sun's rays which produce the shadow.

#### Checking the accuracy of your data

a) How is it possible for you to verify that the angles you measure are correct? This is important in calculating the error in your measurement of the circumference of the earth. Use the relationship between the measured sun angle and your latitude. What is this relationship? At the equinox, the measured sun angle is equal to your latitude. On all other days, the measured sun angle is equal to your latitude minus the declination of the sun. The declination of the sun is the angle the sun makes with the earth's equator. On the first day of sping and fall (the equinox), the declination is zero. On the summer solstice, the declination of the sun is +23.4. On the winter solstice the declination is -23.4. Draw a diagram to verify the relationship between the declination, sun angle, and latitude.

b) How are you going to know what time it is when you are out there measuring sun angles? Because this is a collaborative project, everyone's watches must be synchronized. How sensitive would you expect your sun angle measurements to be to errors in time? It turns out that an accurate measurement of time is much more important for a determination of true north than it is for a measurement of sun angle at local noon. Check that this is true. What changes faster, the length of the shadow or the direction the shadow points? Why is this important?

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