science question?

Question

what is the rate at which the earth's surface is curved per mile? and how is this calculated? any info in reference to these questions will be much appreciated.

Answer 1

You need to refer to basic formulas used in circle geometery. I'm going to use metric to make it simple (after all this is the science section).

Here's what we know:

>the radius R of the earth is about 6360 km
> the arc is 1 km

You need to calculate the distance from the surface of the earth (the circumference) to the mid point of the arc (the sagitta or segment height) to find the difference in height between the earth's surface at the mid point of the arc and where the arc intersects the earth's surface. Then you need to double it because the maximum difference is at the arc's mid point, so you need to allow for the curvature on the other half of the arc as well.

So using the intersecting chord theorem, the answer is 0.0000197 km ( 0.0197 meters) of curvature per half arc, or 0.0394 m per km of full arc.

You can do this in feet and miles if you want, same formulas, and you get 0.332 feet of curvature per mile.

Of course this is all assuming you are dealing with a "flat" surface such as sea level. Unfortunately its not possible to show the circle diagrams here to make it simpler to understand, but any basic geometry text book should show the relationship between the radius, circumference, arc, and sagitta of a circle - and the appropriate formulas.

Answer 2

You're talking about an angle of a circle. The distance across the surface would be the arc length. The angle would be the arc length divided by the radius of the earth. This will give you the answer in radian measure.this link shows you a diagram of what I'm talking about and how to convert radians to degrees.

http://en.wikipedia.org/wiki/Radian_measure