How to Find the Distance Between Two Points on the Surface of a Sphere

Formula

• 1

  Convert the coordinates of the two points to decimal degrees from degrees, minutes and seconds if they are not already in decimal degrees. Multiply the degrees by 1, the minutes by .01667 and the seconds by .0002778. Add all of the results together to get the degrees in decimal degrees. If any of the coordinates are preceded by a S or W, make the coordinate a negative number.

• 2

  Convert each of the degree readings to radians by multiplying by (pi/180).

• 3

  Multiply the sine of the latitude of the first point by the sine of the latitude of the second point.

• 4

  Multiply the cosine of the latitude of the first point by the cosine of the latitude of the second point.

• 5

  Subtract the longitude of the westernmost point from the longitude of the easternmost point. Take the cosine of the absolute value of that result and multiply it by the result of Step 4.

• 6

  Add the result of Step 5 to the result of Step 3. Take the arccosine of the result to get the distance between the two points in radians.

• 7

  Multiply the radians by the radius of the sphere in order to get the actual distance between the two points.

Example

• 8

  Calculate the distance from Miami to Milan as an example. Miami is located at approximately 25°47'16"N 80°13'27"W. Milan is located at approximately 45°27′51″N 09°11′25″E.

• 9

  Convert the measurements to decimal degrees. Miami's latitude is (25*1)+(47*.01667)+(16*.0002778)N = 25.7878. Miami's longitude is (80*1)+(13*.01667)+(27*.0002778)W=-80.2242. Milan's latitude is (45*1)+(27*.01667)+(51*.0002778)N=45.4641. Milan's longitude is (09*1)+(11*.01667)+(25*.0002778)E=9.1903.

• 10

  Convert the degrees to radians. Miami's latitude is 25.7878*(pi/180)=0.45008. Miami's longitude is 80.2242*(pi/180)=-1.40018. Milan's latitude is 45.4641*(pi/180)=0.79350. Milan's longitude is 9.1903*(pi/180)=.16040.

• 11

  Multiply the sine of the latitude of Miami to the sine of the latitude of the Milan: sin(0.45008)*sin(0.79350)=0.3101.

• 12

  Multiply the cosine of the latitude of Miami to the cosine of the latitude of Milan: cos(0.45008)*cos(0.79350)=0.631508466.

• 13

  Subtract the longitude of the Miami from the longitude of Milan. Take the cosine of the absolute value of that result and multiply it by the result of Step 5: cos(abs(-1.40018-.16040))=0.0102161491. 0.0102161491*0.631508466=0.00645158465.

• 14

  Add the result of Step 6 to the result of Step 4. The arccosine of the result is the distance between the two points in radians: 0.00645158465+.3101=0.316551585. acos(0.316551585)=1.24870442.

• 15

  Multiple the final result in radians by the radius of the sphere. In this case, the radius of the Earth is 3963 miles: 1.24870442*3963=4949 miles.