Although it’s slightly flattened at the poles, the Earth is basically a sphere, and on a spherical surface, you can express the distance between two points in terms of both an angle and a linear distance. The conversion is possible because, on a sphere with a radius “r,” a line drawn from the center of the sphere to the circumference, the arc length “L” traced out when the angle changes by “A” number of degrees is:
(L=frac{2pi r A}{360})
Since the radius of the Earth is a known quantity – 6,371 kilometers according to NASA – you can convert directly from L to **A **and vice versa.
How Far Is One Degree?
Converting NASA’s measurement of the Earth’s radius into meters and substituting it in the formula for arc length, we find that each degree the radius line of the Earth sweeps out corresponds to 111,139 meters. If the line sweeps out an angle of 360 degrees, it covers a distance of 40,010, 040 meters. This is a little less than the actual equatorial circumference of the planet, which is 40,030,200 meters. The discrepancy is due to the fact that the Earth bulges at the equator.
Longitudes and Latitudes
Each point on the Earth is defined by unique longitude and latitude measurements, which are expressed as angles. Longitude is the angle between that point and the equator, while latitude is the angle between that point and a line that runs pole-to-pole through Greenwich, England.
If you know the longitudes and latitudes of two points, you can use this information to calculate the distance between them. The calculation is a multistep one, and because it’s based on linear geometry – and the Earth is curved – it’s approximate.
1. Determine the Separation of Latitude
Subtract the smaller latitude from the larger one for places that are both located in the Northern Hemisphere or both in the Southern Hemisphere. Add the latitudes if the places are in different hemispheres.
2. Determine the Separation of Longitude
Subtract the smaller longitude from the larger one for places that are both in the Eastern or both in the Western Hemisphere. Add the longitudes if the places are in different hemispheres.
3. Convert Degrees of Separation to Distances
Multiply the degrees of separation of longitude and latitude by 111,139 to get the corresponding linear distances in meters.
4. Use the Pythagorean Theorem
Consider the line between the two points as the hypotenuse of a right-angled triangle with base “x” equal to the latitude and height “y” equal to the longitude between them. Calculate the distance between them (d) using the Pythagorean theorem:
(d^2=x^2+y^2)
