I’m reading Piers Boziny’s “How to build your own spaceship: The science of personal space travel” ISBN 978-0-452-29533-9.
On page 32 of this paperback, he mentions that the circumference of the earth at the equator is 24860 statute miles. I got to wondering how to figure out the circumference of the earth at different latitudes. I also got to wondering if I could use some stuff I just learned in calculus to develop a general formula. Turns out I was able to.
The first step is to assume the earth is a perfect sphere and calculate its radius at the equator. Circumference = 2 times pi times radius, so 24860 = 2 times pi times radius, so 24860 divided by 2 divided by pi = radius, so radius = 3957 statute miles.
The second step is to figure out the formula for the horizontal radius at any given point. To do so, graph yourself a circle centered at the origin (0,0) with a radius of 1 unit. Then shoot a line out to the upper right from the origin at some angle theta from the x-axis. The point where that line intersects your circle is the point (cos theta, sin theta). To calculate the horizontal circumference on this ideal circle of 1 unit at any “latitude” theta, the formula is: Circumference at “latitude” theta = 2 times pi times cos theta.
To correct this formula for the fact that we’re dealing with a sphere having a radius of 3957 statute miles instead of 1 unit, let’s correct the formula: East-west circumference of the earth at latitude theta = 2 times pi times cos theta times 3957 statute miles.
Therefore, as an example, the distance around the world at 40°N latitude is 2 times pi times cos theta times 3957 miles, or 2 times pi times cos 40° times 3957 miles, or 2 times pi times 0.7660 times 3957 miles, or 19046 statute miles east-west at 40°N latitude.
Isn’t that fun?
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