Convert your origin's coordinates to decimal form. Typically, you write latitude and longitude coordinates using the degrees-minutes-seconds (DMS) format. For example, Miami, Florida's coordinates are 25° 46' 27 N, 80° 11' 37 W. Convert "minutes" (the second group of numbers in a coordinate) to degrees by diving the number of minutes by 60---there are 60 "minutes" in a degree. For Miami's latitude and longitude, this produces 46/60 and 11/60, or .7667 and .1833, respectively. As in the case with time, there are 60 "seconds" in every minute. Keeping in mind that there are also 60 minutes in each degree, divide the number in seconds by 60, then by 60 again---or simply by 3,600 in the first place. For Miami's latitude and longitude, this would be 27/3,600 and 37/3,600, or .0075 and .0103. Add your degrees, minutes and seconds to get 25.7742, --80.1936, keeping in mind that "W" and "S" notations cause a number to be negative.
Convert your destination's coordinates as well. If your destination is Rio de Janeiro, Brazil, coordinates 22° 53' 60 S, 43° 13' 60 W, calculate its decimal coordinates in the same way. Notice that 53/60 and 13/60 equal .8833 and .2167, respectively, and that 60/3,600 (the same for each seconds coordinate) equals .0167. Add these numbers together to find that Rio de Janeiro's coordinates are -22.9000, --43.2334.
Plug your coordinates into the equation "t = Arccos[Cos[a1] Cos[b1] Cos[a2] Cos[b2] + Cos[a1] Sin[b1] Cos[a2] Sin[b2] + Sin[a1] Sin[a2]]/360 * 2π * r, where "t" is the distance between your points in kilometers, "π" is pi (3.14159) and "r" is the radius of the earth, or 6,378.1 km, "a" and "b" represent latitude and longitude, respectively, and "1" and "2" represent your origin and destination. For the example, note that t =
Arccos[Cos[25.7742]Cos[-80.1936]Cos[-22.9000]Cos[-43.2334] + [Cos[25.7742]Sin[-80.1936]Cos[-22.9000]Sin[-43.2334] + Sin[25.7742]Sin[-22.9000]]/360 * 2π * 6,378.1 = Arccos [(.9005 x .1703 x .9212 x .7286) + (.9005 x -.9854 x .9212 x -.6850) + (.4348 x -.3891)]/360 * 40,074.8 = Arccos[.1029 + .5599 - .1692]/360 * 40,074.8 = Arccos [.4936] /360 * 40,074.8 = 60.4225/360 * 40,074.8 = .1678 * 40,074.8 = 6,726 km of distance between Miami and Rio de Janeiro.
Check your results by plugging your coordinates into Movable Type Scripts' "Distance Calculator," located in the Resources section. For Miami--Rio de Janeiro, it gives 6,718 km, a 8 km (or a .01% error) you can attribute to rounding.
