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Geographic coordinate system
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==Length of a degree== {{Main|Length of a degree of latitude|Length of a degree of longitude}} {{See also|Arc length#Great circles on Earth}} On the [[Geodetic Reference System 1980|GRS{{nbsp}}80]] or [[World Geodetic System#WGS84|WGS{{nbsp}}84]] spheroid at [[sea level]] at the Equator, one latitudinal second measures 30.715 [[metre|m]], one latitudinal minute is 1843 m and one latitudinal degree is 110.6 km. The circles of longitude, meridians, meet at the geographical poles, with the west–east width of a second naturally decreasing as latitude increases. On the [[Equator]] at sea level, one longitudinal second measures 30.92 m, a longitudinal minute is 1855 m and a longitudinal degree is 111.3 km. At 30° a longitudinal second is 26.76 m, at Greenwich (51°28′38″N) 19.22 m, and at 60° it is 15.42 m. On the WGS{{nbsp}}84 spheroid, the length in meters of a degree of latitude at latitude {{mvar|ϕ}} (that is, the number of meters you would have to travel along a north–south line to move 1 degree in latitude, when at latitude {{mvar|ϕ}}), is about {{block indent|1= <math>111132.92 - 559.82\, \cos 2\phi + 1.175\, \cos 4\phi - 0.0023\, \cos 6\phi</math><ref name=GISS>[http://gis.stackexchange.com/questions/75528/length-of-a-degree-where-do-the-terms-in-this-formula-come-from] {{Webarchive|url=https://web.archive.org/web/20160629203521/http://gis.stackexchange.com/questions/75528/length-of-a-degree-where-do-the-terms-in-this-formula-come-from |date=29 June 2016 }} Geographic Information Systems – Stackexchange</ref> }} The returned measure of meters per degree latitude varies continuously with latitude. Similarly, the length in meters of a degree of longitude can be calculated as {{block indent|1= <math>111412.84\, \cos \phi - 93.5\, \cos 3\phi + 0.118\, \cos 5\phi</math><ref name=GISS/> }} (Those coefficients can be improved, but as they stand the distance they give is correct within a centimeter.) The formulae both return units of meters per degree. An alternative method to estimate the length of a longitudinal degree at latitude <math>\phi</math> is to assume a spherical Earth (to get the width per minute and second, divide by 60 and 3600, respectively): {{block indent|1= <math> \frac{\pi}{180}M_r\cos \phi \!</math> }} where [[Earth radius#Meridional Earth radius|Earth's average meridional radius]] <math>\textstyle{M_r}\,\!</math> is {{nowrap|6,367,449 m}}. Since the Earth is an [[Spheroid#Oblate spheroids|oblate spheroid]], not spherical, that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude <math>\phi</math> is {{block indent|1= <math>\frac{\pi}{180}a \cos \beta \,\!</math> }} where Earth's equatorial radius <math>a</math> equals 6,378,137 m and <math>\textstyle{\tan \beta = \frac{b}{a}\tan\phi}\,\!</math>; for the GRS{{nbsp}}80 and WGS{{nbsp}}84 spheroids, <math display="inline">\tfrac{b}{a}=0.99664719</math>. (<math>\textstyle{\beta}\,\!</math> is known as the [[Latitude#Parametric (or reduced) latitude|reduced (or parametric) latitude]]). Aside from rounding, this is the exact distance along a parallel of latitude; getting the distance along the shortest route will be more work, but those two distances are always within 0.6 m of each other if the two points are one degree of longitude apart. {| class="wikitable" |+ Longitudinal length equivalents at selected latitudes |- ! style="width:100px;" | Latitude ! style="width:150px;" | City ! style="width:100px;" | Degree ! style="width:100px;" | Minute ! style="width:100px;" | Second ! style="width:100px;" | 0.0001° |- | 60° | [[Saint Petersburg]] | style="text-align:center;" | 55.80 km | style="text-align:center;" | 0.930 km | style="text-align:center;" | 15.50 m | style="text-align:center;" | 5.58 m |- | 51° 28′ 38″ N | [[Greenwich]] | style="text-align:center;" | 69.47 km | style="text-align:center;" | 1.158 km | style="text-align:center;" | 19.30 m | style="text-align:center;" | 6.95 m |- | 45° | [[Bordeaux]] | style="text-align:center;" | 78.85 km | style="text-align:center;" | 1.31 km | style="text-align:center;" | 21.90 m | style="text-align:center;" | 7.89 m |- | 30° | [[New Orleans]] | style="text-align:center;" | 96.49 km | style="text-align:center;" | 1.61 km | style="text-align:center;" | 26.80 m | style="text-align:center;" | 9.65 m |- | 0° | [[Quito]] | style="text-align:center;" | 111.3 km | style="text-align:center;" | 1.855 km | style="text-align:center;" | 30.92 m | style="text-align:center;" | 11.13 m |} <!--The Equator is the [[fundamental plane (spherical coordinates)|fundamental plane]] of all geographic coordinate systems. All spherical coordinate systems define such a fundamental plane.-->
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