3

I have a question, how to calculate distance from athens to san francisco?

here is what I did, but it is different than the answer I searched on google

San Francisco & Athena both have 38 degree North Latitude;

San Francisco has 122 degree West Longitude;

Athena has 24 degree East Longitude

90 degree - 38 degree = 52 degree

sin(52) = opp/hyp= AthensX/R = AthensX/3960

AthensX = 3960Sin(52) = (3960)(.7880) = 3120.522584

122 degree - -24 =degree 146 degree or 2.5481 rad

s = rθ = (3120.522584)(2.5481) = 7951.655447 mile

if don't understand, see the picture below

https://i.stack.imgur.com/hhOXq.png (sorry for my ugly hand writing)

The distance I searched on google is 6783.616 mile

I have no idea about it X~X

Rex Rau
  • 85
  • 1
    I'm not entirely sure if I can follow what you're doing. Could you perhaps more focus on the steps you've taken than on the actual calculation?

    As far as it's possible to judge yet, there seem to be two mistakes: first of all, why do you subtract 24 from 122 instead of adding those? Secondly, you seem to assume the distance from SF to Athens is the length of the route along the $38^\circ$ North Latitude. This is not actually the case, however. Shortest paths on spheres are parts of the circle through the points having the same origin as the sphere.

     – HSN Apr 30 '13 at 10:05
  • Thank for you reply, I updated, if it still look bad, tell me I will implement it again. And to the first mistake you said, I subtract 122 from 24 because I am finding the length between Athens and SF, so to be able to get the length, I need to get the angle subtend to. – Rex Rau Apr 30 '13 at 10:31
  • Honestly, I am not sure about the logic of the calculation. I calculated this by following an example in a math book. – Rex Rau Apr 30 '13 at 10:33
  • 1
    For some basic information about writing math at this site see e.g. here. – Américo Tavares Apr 30 '13 at 10:37
  • To continue on what @HSN said: SF is 122 degree West while Athens is 24 degree East. The difference in directions is a difference in a sign, that is, if the meridian is 0 and east is positive, SF is at $-122^\circ$ and Athens at $+24^\circ$. Do you see now why you should add instead of subtract? – Willie Wong Apr 30 '13 at 10:42
  • @RexRau: It is $24$ East and $122$ West. Like the distance betweem $34$ and $-122$. – André Nicolas Apr 30 '13 at 10:42
  • oh, ok, but even if I added, when i use s = rθ = (3120.522584)(2.5481), I get 7951.655447 mile, it is still not the same as what I searched on google 6783.616 mile X~X. And thank you for your replies – Rex Rau Apr 30 '13 at 10:48
  • 1
    @Rex: and there's the second problem as HSN mentioned: you've computed the distance along the 38 degree parallel, not the "great circle distance", which would be in general shorter. – Willie Wong Apr 30 '13 at 10:50
  • so the 6783.616 is from the great circle distance, and how about my 7951.655447? it is correct? – Rex Rau May 01 '13 at 02:18

1 Answers1

1

First you should go read this answer about the spherical laws of cosines. This will tell you how to compute the "angle" between two points given the latitude and longitude.

Then you multiply the angle, in radians, against the radius of the earth to get the distance in a unit of length.

Another very good explanation is given by our own John Cook on his website.

Community
  • 1
Willie Wong
  • 73,139