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Distance Calculation

Any point on the Earth’s surface can be located by its latitude and longitude coordinates. Latitude is the angle above or below the equator in degrees. The equator is zero degrees, the North pole is north 90 degrees latitude, and the South pole is South 90 degrees latitude. The continental United States falls between 25 and 50 degrees North.

Longitude is the angle East or West of the Greenwich meridian. The continental United States is between 70 and 125 degrees West.

Approximate Solutions

One degree of latitude is equal to 69.1 miles. One degree of longitude is equal to 69.1 miles at the equator. North or South of the equator, 1 degree of longitude is a smaller distance. It’s reduced by the cosine of the latitude. Dividing the latitude number by 57.3 converts it to radians.

\[DistanceLat = 69.1\, (Lat_2 - Lat_1)\]

\[\begin{split}DistanceLong = 69.1\, (Long_2- Long_1)\, cos\, (\frac{Lat_1}{57.3})\\\end{split}\]

\[Distance = \sqrt{DistanceLat^2 + DistanceLong^2}\]

If you don’t want to use the COS function, then a good approximate solution is:

\[DistanceLat = 69.1\, (Lat_2 - Lat_1)\]

\[DistanceLong = 53\, (Long_2 - Long_1)\]

\[Distance = \sqrt{DistanceLat^2 + DistanceLong^2}\]

Exact Solution

To calculate the exact distance between points requires spherical geometry.

The basic formula is:

\[D = 3959\, cos^{-1}\, (sin\, (Lat_1)\, sin(Lat_2)\, + cos\, (Lat_1)\, cos\, (Lat_2)\, cos\, (Long_2 - Long_1))\]

The above formula has one major problem: most computer languages do not support the arccosine function. Therefore a different form is necessary, one that uses the arctangent function that most languages embrace.

\[\begin{split}D = 3959\, tan^{-1}\, (\frac{\sqrt{1-A^2}}{A})\\\end{split}\]

where A is equal to:

\[A = sin\, (Lat_1)\, sin\, (Lat_2) + cos\, (Lat_1)\, cos\, (Lat_2)\, cos\, (Long_2-Long_1)\]

Most computer languages compute the sine and cosine function with the angle given in radians. To convert degrees to radians, divide degrees by the constant:

\[\begin{split}57.3\,({\frac{180}{\pi}\\})\end{split}\]

In BASIC this would be programmed as follows:

\[C = 57.3\]

\[\begin{split}A = sin\, ({\frac{Lat_1}{C})\\}\, sin\, ({\frac{Lat_2}{C})\\} + cos\, ({\frac{Lat_1}{C}\\})\, cos\, ({\frac{Lat_2}{C}\\})\, cos\, ({\frac{Long_2}{C}\\}-{\frac{Long_1}{C}\\})\end{split}\]

\[\begin{split}D = 3959\, tan^{-1}\, (\frac{\sqrt{1-A^2}}{A})\\\end{split}\]

Where:

D = distance in statue miles from the first to the second point.

C = degrees to radians constant 57.3.

Lat₁ = latitude of the first point in degrees.

Long₁ = longitude of the first point in degrees.

Lat₂ = latitude of the second point in degrees.

Long₂ = longitude of the second point in degrees.

NOTE: Dividing the latitude number by 57.3 converts it to radians.

Since all ZIP Code points in the United States are North latitude and West longitude, there is no need to check the sines (positive and negative) of the latitudes and longitudes.

Some programming languages, such as dBASE II/III, do not have the functions of the sine, cosine and arctangent. Also, the formulas given are time-consuming to calculate. A simpler but less accurate method is given here:

\[D = 69.1\, \sqrt{(Lat_2-Lat_1)^{2} + 0.6\, (Long_2 - Long_1)^{2}}\]

Although this formula is not as accurate as the first great circle method, it will give good results for most applications.

In basic, it is written as follows:

\[D = 69.1\, \sqrt{(Lat_2-Lat_1)^{2} + .6\, (Long_2 - Long_1)^{2}}\]

In dBASE III, it is written as follows:

\[D = 69.1\, \sqrt{(Lat_2-Lat_1)^{2} + 0.6\, (Long_2 - Long_1)^{2}}\]

To make using the database easier, the latitudes and longitudes are given in the decimal format instead of the degree, minute and second format. The latter would require the additional steps of converting seconds to fractions of a minute and then minutes to a fraction of a degree.

Bearing Formula

The bearing is the direction from the first point to the second point. It is expressed as an angle from North in degrees. Due North is a bearing of zero degrees, East is a bearing of 90 degrees, South is 180, and west is 270.

The bearing from the first point to the second is calculated with:

\[B = tan^{-1}\, (\frac{sin(Lat_1)\, sin(Long_2 - Long_1)}{sin(Lat_2)\, cos(Lat_1) - cos(Lat_2)\, sin(Lat_1)\, cos(Long_2 - Long_1)})\]

Since the ARCTAN function gives the angle in radians, it is necessary to convert B to degrees by multiplying by

\[\begin{split}57.3\,({\frac{180}{\pi}\\})\end{split}\]

To use this equation requires some special considerations. This example assumes that Lat₁, Lat₂, Long₁, Long₂ have been converted to radians.

Use the following steps:

\[N = sin\, (Lat_1)\, sin\, (Long_1 - Long_2)\]

\[D = sin\, (Lat_2)\, cos\, (Lat_1)\, – \, cos\, (Lat_2)\, sin\, (Lat_1)\, cos\, (Long_1 - Long_2)\]

\[\begin{split}B = 57.3\, tan^{-1}\, ({\frac{N}{D}\\})\end{split}\]

If (D > 0) then (B = 360 + B)

If (D < 0) then (B = 180 + B)

If (B < 0) then (B = 360 + B)

Where:

N is the Numerator.

D is the Denominator.

B is the Bearing.

Examples

The following examples should give you a feel for the numbers. For these examples, we will use the points of Schenectady, NY and Los Angeles, CA.

First Point is in Schenectady, NY, 12345

Lat₁ = 42.8145

Long₁ = 73.9380

Second Point is in Los Angeles, CA, 90001

Lat₁ = 34.0515

Long₁ = 118.2420

Example 1: Approximate Formula

\[DistanceLat = 69.1\, (Lat_2 - Lat_1) = 605.5\]

\[DistanceLong = 69.1\, (Long_2- Long_1)\, cos\, (\frac{42.8145}{57.3}) \ = 2245.8\]

\[Distance = 2326.0 \ miles\]

Example 2: Approximate Formula

\[DistanceLat = 69.1\, (Lat_2 - Lat_1) = 605.5\]

\[DistanceLong = 53\, (Long_2 - Long_1) = \ 2348.1\]

\[Distance = 2424.9 \ miles\]

Example 3: Exact Solution Formula

\[\begin{split}A_1 = sin\, ({\frac{Lat_1}{57.3})\\}\, sin\, ({\frac{Lat_2}{57.3})\\} = \ 0.3805\end{split}\]

\[\begin{split}A_2 = cos\, ({\frac{Lat_1}{57.3}\\})\, cos\, ({\frac{Lat_2}{57.3}\\})\, cos\, ({\frac{Long_2}{57.3}\\}-{\frac{Long_1}{57.3}\\}) = \ 0.4350\end{split}\]

\[A =\ A_1 + A_2 = \ 0.8155\]

\[\begin{split}D = 3959\, tan^{-1}\, (\frac{\sqrt{1-A^2}}{A})\\\end{split}\]

\[D = 3959\, tan^{-1}\, (\frac{\sqrt{1-(0.8155)^2}}{0.8155}) \ = 2443.4\]

Example 4: Bearing Formula

\[\begin{split}N = sin\, ({\frac{Lat_1}{57.3})\\}\, sin\, ({\frac{Long_2}{57.3}\\}-{\frac{Long_1}{57.3}\\}) = \ 0.4747\end{split}\]

\[\begin{split}D_1 = sin\, ({\frac{Lat_2}{57.3})\\}\, cos\, ({\frac{Lat_1}{57.3})\\} = \ 0.4107\end{split}\]

\[\begin{split}D_2 = cos\, ({\frac{Lat_2}{57.3})\\}\, sin\, ({\frac{Lat_1}{57.3})\\}\, cos\, ({\frac{Long_2}{57.3}\\}-{\frac{Long_1}{57.3}\\}) = \ 0.4029\end{split}\]

\[\begin{split}B = 57.3\, tan^{-1}\, ({\frac{N}{D_1 - D_2}\\}) = \ -81.9 \,Degrees\end{split}\]

\[Bearing = 360 + B = 360 - 89.1\]

\[Bearing = 270.9\, degrees\, from\, North\]

LATAs

United States LATAs

Code	State	Code	State
120	Maine	222	Delaware Valley, NJ
122	New Hampshire	224	North Jersey, NJ
124	Vermont	226	Capital, PA
126	Western Mass	228	Philadelphia, PA
128	Eastern Mass	230	Altoona, PA
130	Rhode Island	232	Northeast, PA
132	New York, NY	234	Pittsburgh, PA
133	Poughkeepsie, NY	236	Washington DC
134	Albany, NY	238	Baltimore, MD
136	Syracuse, NY	240	Hagerstown, MD
138	Binghamton, NY	242	Salisbury, MD
140	Buffalo, NY	244	Roanoke, VA
220	Atlantic Coastal, NJ	246	Culpeper, VA
248	Richmond, VA	428	Wilmington, NC
250	Lynchburg, VA	430	Greenville, SC
252	Norfolk, VA	432	Florence, SC
254	Charleston, WV	434	Columbia, SC
256	Clarksburg, WV	436	Charleston, SC
320	Cleveland, OH	438	Atlanta, GA
322	Youngstown, OH	440	Savannah, GA
324	Columbus, OH	442	Augusta, GA
325	Akron, OH	444	Albany, GA
326	Toledo, OH	446	Macon, GA
328	Dayton, OH	448	Pensacola, FL
330	Evansville, IN	450	Panama City, FL
332	South Bend, IN	452	Jacksonville, FL
334	Auburn/Huntington, IN	454	Gainesville, FL
336	Indianapolis, IN	456	Daytona Beach, FL
338	Bloomington, IN	458	Orlando, FL
340	Detroit, MI	460	Southeast FL
342	Upper Peninsula MI	462	Louisville, KY
344	Saginaw, MI	464	Owensboro, KY
346	Lansing, MI	466	Winchester, KY
348	Grand Rapids, MI	468	Memphis, TN
350	Northeast WI	470	Nashville, TN
352	Northwest WI	472	Chattanooga, TN
354	Southwest WI	474	Knoxville, TN
356	Southeast WI	476	Birmingham, AL
358	Chicago, IL	477	Huntsville, AL
360	Rockford, IL	478	Montgomery, AL
362	Cairo, IL	480	Mobile, AL
364	Sterling, IL	482	Jackson, MS
366	Forrest, IL	484	Biloxi, MS
368	Peoria, IL	486	Shreveport, LA
370	Champaign, IL	488	Lafayette, LA
374	Springfield, IL	490	New Orleans, LA
376	Quincy, IL	492	Baton Rouge, LA
420	Asheville, NC	520	St Louis, MO
422	Charlotte, NC	521	Westphalia, MO
424	Greensboro, NC	522	Springfield, MO
426	Raleigh, NC	524	Kansas City, MO
526	Fort Smith, AK	652	Idaho
528	Little Rock, AK	654	Wyoming
530	Pine Bluff, AK	656	Denver, CO
532	Wichita, KS	658	Colorado Springs, CO
534	Topeka, KS	660	Utah
536	Oklahoma City, OK	664	New Mexico
538	Tulsa, OK	666	Phoenix, AZ
540	El Paso, TX	668	Tucson, AZ
542	Midland, TX	670	Eugene, OR
544	Lubbock, TX	672	Portland, OR
546	Amarillo, TX	674	Seattle, WA
548	Wichita Falls, TX	676	Spokane, WA
550	Abilene, TX	720	Reno, NV
552	Dallas, TX	721	Pahrump, NV
554	Longview, TX	722	San Francisco, CA
556	Waco, TX	724	Chico, CA
558	Austin, TX	726	Sacramento, CA
560	Houston, TX	728	Fresno, CA
562	Beaumont, TX	730	Los Angeles, CA
564	Corpus Christi, TX	732	San Diego, CA
566	San Antonio, TX	734	Bakersfield, CA
568	Brownsville, TX	736	Monterey, CA
570	Hearne, TX	738	Stockton, CA
620	Rochester, MN	740	San Luis Obispo, CA
624	Duluth, MN	832	Alaska
626	St Cloud, MN	834	Hawaii
628	Minneapolis, MN	920	Connecticut
630	Sioux City, IA	921	Fishers Island, NY
632	Des Moines, IA	922	Cincinnati, OH
634	Davenport, IA	923	Lima/Mansfield, OH
635	Cedar Rapids, IA	924	Erie, PA
636	Brainerd/Fargo, ND	927	Harrisonburg, VA
638	Bismark, ND	928	Charlottesville, VA
640	South Dakota	929	Eninburg, VA
644	Omaha, NE	932	Blue Field, WV
646	Grand Island, NE	937	Richmond, IN
648	Great Falls, MT	938	Terre Haute, IN
650	Billings, MT	939	Fort Myers, FL
949	Fayetteville, NC	963	Kalispell, MT
951	Rocky Mount,	973	Palm Springs, CA
952	Tampa, FL	974	Rochester, NY
953	Tallahassee, FL	976	Mattoon, IL
956	Bristol/Johnson, TN	977	Macomb, IL
958	Lincoln, NE	978	Olney, IL
960	Couer D’Alene, ID	980	Navajo Territory, AZ
961	San Angelo, TX	981	Navajo Territory, UT

Canadian LATAs

Code	State
850	Quebec
851	Ontario
881	Alberta
883	Quebec
884	Alberta
885	Newfoundland
886	British Columbia
887	Nova Scotia/Prince Edward
888	Manitoba
889	Nova Scotia/Prince Edward
890	New Brunswick
891	Saskatchewan
892	Yukon

Caribbean LATAs

Code	State
820	Puerto Rico
822	Virgin Islands
824	Bahamas
826	Jamaica
828	Dominican Republic
830	Other Caribbean Islands

Additional LATAs

Code	State
838	Mexico
870	Mariana Islands
871	Guam

State Abbreviations and Codes

Abbreviation	Code	State
AL	01	Alabama
AK	02	Alaska
AZ	04	Arizona
AR	05	Arkansas
CA	06	California
CO	08	Colorado
CT	09	Connecticut
DE	10	Delaware
DC	11	District of Columbia
FL	12	Florida
GA	13	Georgia
HI	15	Hawaii
ID	16	Idaho
IL	17	Illinois
IN	18	Indiana
IA	19	Iowa
KS	20	Kansas
KY	21	Kentucky
LA	22	Louisiana
ME	23	Maine
MD	24	Maryland
MA	25	Massachusetts
MI	26	Michigan
MN	27	Minnesota
MS	28	Mississippi
MO	29	Missouri
MT	30	Montana
NE	31	Nebraska
NV	32	Nevada
NH	33	New Hampshire
NJ	34	New Jersey
NM	35	New Mexico
NY	36	New York
NC	37	North Carolina
ND	38	North Dakota
OH	39	Ohio
OK	40	Oklahoma
OR	41	Oregon
PA	42	Pennsylvania
RI	44	Rhode Island
SC	45	South Carolina
SD	46	South Dakota
TN	47	Tennessee
TX	48	Texas
UT	49	Utah
VT	50	Vermont
VA	51	Virginia
WA	53	Washington
WV	54	West Virginia
WI	55	Wisconsin
WY	56	Wyoming
AS	60	American Samoa
FM	64	Micronesia
GU	66	Guam
MH	68	Marshall Islands
MP	69	N.Mariana Islands
PW	70	Palau
PR	72	Puerto Rico
UM	74	Minor Islands
VI	78	Virgin Islands

Canadian Abbreviations and Provinces

Abbreviation	Province
BC	British Columbia
NB	New Brunswick
NL	Newfoundland
NS	Nova Scotia
NT	Northwest Territories
NU	Nunavut
PE	Prince Edward Island
AB	Province of Alberta
MB	Province of Manitoba
ON	Province of Ontario
QC	Province of Quebec
SK	Province of Saskatchewan
YT	Yukon Territories

Release Schedule

Release Date Schedule
2024-10-21
2025-01-27
2025-04-21
2025-07-21
2025-10-27