What
this
means is that if you can measure the sun's angles at two different
positions
on the earth at the same time, you can figure out the central angle!
Let's
look at an example to see how this information leads to determining an
empirical value for the circumference of the earth.
Our two sites will be at Manasquan, New Jersey and San Juan, Puerto Rico. |
In Manasquan this angle
would be a
little less than 39 degrees.
In San Juan the angle was about 18.5 degrees. |
At
local
noon on the same day, the experimenters measured the shadow cast by
a meter stick.
The central angle equals the
difference
between these angles. The Geometer's Sketchpad can be used to get more
accurate measurement for the angles.
Using these measurements, the central angle is 38.85 - 18.59 or 20.26 degrees. Now that we know the central angle, we can determine how many such angles would make up the full circle. That is, how many "slices" can we make where the angle is 20.26 degrees? Since the total is 360 degrees, then you would have a little less than 18 equal "slices" (or 17.77 to be more precise.) Each slice's edge represents the distance between Manasquan and Puerto Rico, or more accurately, the lateral distance between the two sites. But why the lateral distance and
not just
the direct distance? (more about this later.)
So if the length of the "slice" is about 2400 km and there are about 18 slices then the projected circumference is 2400 x 18 = 43,200 km which is about an 8% error since the average circumference is 40,008 kms. (More precisely, it is 2404 km times 17.7 "slices".) |
Method
2: Using Trigonometry
Skip
the trig. Take me to the "How to do the experiment" pages