Astronomy 101-1040

Converting shadow lengths into degrees

Once you have your observation (and the New Mexico observation), you must convert each into degrees. If you are familiar and comfortable with trigonometry, the equation is:

tan A = H/L

where "A" is the angle in question, H is the height of the gnomon in whatever unit you are using (such as mm, cm or inches), and "L" is the shadow length in the same units. Again, remember that the "/" means "divided by." This can also be written:

A = arctan(H/L)

Be sure that H and L are in the same units, and that your calculator is set for degrees rather than radians.

ADDITIONAL BACKGROUND: Since the Earth is round (roughly spherical), the Sun's angle will change depending on where you make the observation. If on the same day two observers in different locations on a north-south line observe the angle of the Sun when it is highest in the sky, then the difference in angle (call it "Delta") plus the actual distance (call it "X") between the observers can be used to set up the following ratio:

Delta degrees/360 degrees = X/Circumference of Earth

Read this as "Delta degrees divided by 360 degrees equals the distance X divided by the circumference of the Earth."

You may realize that this is an equation. The 360 degrees is simply the number of degrees all around the Earth. If you actually measure Delta and X, then all that is left is the Circumference of the Earth (call it "C"), which is what we want to find. You can rearrange the equation to this:

C = (360/Delta) * X

(The "degrees," being on both top and bottom of the division part (signified by "/") cancel out, so you can ignore them).

Thus to find the circumference of Earth after have found Delta and X, all you need to do is divide 360 by Delta, then multiply the whole thing by X. The result will be C, the circumference of Earth.

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