# How to calculate arc distance on a sphere

I hope my question makes sense. I just don't know how to describe it using math lingo. Please bear with me.

Let's say on a globe I'm traveling from point A to point B that is exactly opposite side of the globe (12,500 miles). Theoratically I can take any path I want (North, South, East, West) and arrive at my destination at the same time and distance. Assuming I'm traveling a straight arc line to my destination. Everyone agrees?

Now, If my destination is not quiet half way around the world, let's say the distance is 10,000 miles away exactly south from me. I cannot just fly any direction I want and arrive to my destination at the same time and distance. But theoratically there should still be a straight line arc with a degree in deviation from my southern destination. I should be able to go straight south, or a few degrees south east or a little bit south west and still make it to my destination and not gain any distance and time.

The shorter the distance becomes, the less deviation in the degree of the arc becomes.

Is there a formula to calculate how much degrees I can deviate with distance around a globe at set distances? Earth's circumference is 25,000 miles.

• should say bear with me. Bare means to take off your clothes – Will Jagy Nov 5 '18 at 22:37
• Geodesics (curves of shortest length) on a sphere are arcs of great circles (i.e. circles that cut the sphere exactly in half). Does that answer your question? – NickD Nov 5 '18 at 22:43
• Take rays from the center of the earth, one ray to your starting point, another ray to your destination. The distance you travel is proportional to the angle between these rays. When the starting and destination point are on opposite sides of the globe, then this angle is 180 degrees, and the distance is you say 12,500 miles. So, if the angle between the rays is 5 degrees, for example, then the distance is proportional to 12,500 miles, with proportionality constant 5/180, so distance is (5/180)*12,500 = about 347 miles – Mirko Nov 5 '18 at 23:02