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### Distance using Latitude and Longitude

```
Date: 12/31/97 at 11:28:30
From: Brian K. Reed
Subject: Distance using Lat and Long ?

Hi!

Is there a simple formula for calculating distance using known
latitude and longitude?

Brian Reed
Senior Research Analyst
Booz*Allen & Hamilton
```

```
Date: 12/31/97 at 11:35:28
From: Doctor Rob
Subject: Re: Distance using Lat and Long ?

Yes.

Assume the Earth is a perfect sphere.  Let all angles be measured in
degrees.

If the latitude is North, let phi = 90 - latitude. If the latitude is
South, let phi = 90 + latitude. The North Pole has phi = 0, the South
Pole has phi = 180, and 0 <= phi <= 180.

If the longitude is East, let theta = longitude. If the longitude is
West, let theta = -longitude. Greenwich, England, has theta = 0, and
-180 <= theta <= 180.

Let the angles for the two points be (phi1, theta1) and (phi2,
theta2). Then compute

c = sin(phi1)*sin(phi2)*cos(theta1-theta2) + cos(phi1)*cos(phi2).

Then the shortest great circle distance between the two points is

d = R*Arccos(c)*Pi/180,

where R is the radius of the earth in miles, and the arccosine is
taken between 0 and 180 degrees, inclusive.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 12/31/97 at 11:56:20
From: Reed Brian
Subject: RE: Distance using Lat and Long ?

Thanks!  This is just what I need!

Regards,
Brian
```

```
Date: 11/16/98 at 12:45:35
From: Tom Reilly
Subject: Wrong formula

The answer from Dr. Rob to a student's question posting [see below]
has an incorrect formula.

c = sin(phi1)*sin(phi2)*cos(theta1-theta2) + cos(phi1)*cos(phi2).

should be:

c = sin(phi1)*sin(phi2)+cos(theta1-theta2) * cos(phi1)*cos(phi2).
```

```
Date: 11/16/98 at 15:34:28
From: Doctor Rick
Subject: Re: Wrong formula

Hi, Tom. Did you notice the way Dr. Rob defined phi? If phi is simply
the latitude (+ for north, - for south latitude), then your formula is
correct - it is the formula I use. But  Dr. Rob defined phi as 90 - lat for north latitudes (90 + lat for south latitudes). This definition switches sines and cosines, since sin(90-phi) = cos(phi)
and cos(90-phi) = sin(phi).

Dr. Rob defined phi this way because it is the standard way that
spherical coordinates are defined. Admittedly it's a little confusing
to those who are used to working with latitudes, but then, those who
are NOT accustomed to latitude math are more likely to ask the question!

Mistakes do sometimes make it into our archives, and we appreciate
diligent readers who catch these errors. But I don't think we are
wrong here. Keep looking! :-)

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/

```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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