The greatcircle distance is the shortest distance between any two points on the surface of a sphere milage calculator measured along a path on the surface of the sphere (as opposed to going through the sphere's interior). Because spherical geometry is rather mileage calulator different from ordinary Euclidean geometry, the equations for distance take on a different mileage calculater form. The distance between two points in Euclidean mileage claculator space is the length of a straight line from one point to the other. mileage caculator On the sphere, however, there mileagecalculator are no straight lines. In nonEuclidean geometry, straight lines are replaced with geodesics. Geodesics on the sphere are the great circles (circles on the sphere mileage calculator whose centers are coincident with the center of the sphere).
Between any two points on gas mileage calculator a sphere which are not fuel mileage calculator directly opposite each other, there is a unique great circle. The two points separate the great circle into two arcs. trip mileage calculator The length of the shorter arc is the greatcircle distance between the points. driving mileage calculator Between two points which mileage calculators are directly opposite each other (called antipodal points) there are infinitely many great circles, but mileage calculators mileage calculators all have the same length, equal to half the circumference of the travel distance mileage milage calculator circle, or πr, where r is the radius of the sphere.
Because the Earth is approximately spherical (see spherical Earth), the equations gas mileage calculator ford explorer for greatcircle distance are important for finding the shortest distance between points car fuel mileage calculator on the mileage calculator for us to mexico surface of the Earth, and so have important applications in navigation.
Contents
 1 The formula
 2 Spherical mileage calculator travel distance on the uk mileage calculator Earth
 3 A worked example
 4 Spherical coordinates
 5 See also
 6 External walking mileage calculator links

The formula
Let be the latitude and longitude of two points, respectively, the longitude difference and the angular difference/distance, or central fuel mileage calculator km angle, which can be constituted from the spherical law of cosines:
This arccosine form can world mileage calculator have large rounding errors for the common case where the car trip gas mileage calculator distance is small, however, so it is not normally metric fuel mileage calculator used. Instead, a simpler equation known historically as the haversine formula was preferred, which is mileage and miles per gallon calculator much more accurate for small distances:
(Historically, the use of this formula was simplified by the availability of tables mileage calculator new brunswick for the haversine function: hav(θ) = sin^{2}(θ/2).) Although this formula is driving mileage calculator between us and mexico accurate for most distances, it too suffers from rounding errors for the special (and somewhat unusual) fuel economy mileage calculator gas case of antipodal points (on opposite ends of the sphere). A more complicated gas mileage calculator by car formula that is accurate for all distances is:^{citation needed]}
(On a computer, one should use atan2 canadian mileage calculator rather than the ordinary arctangent function, in order to correctly handle the case car mileage fuel calculator exel where the denominator is zero.)
If r is fuel mileage conversion calculator the greatcircle radius of the sphere, then the greatcircle distance is .
Note: above, fuel mileage diesel improve get improves tips calculator accuracy refers to rounding errors only; all formulas themselves are exact (for a sphere).
Spherical distance on the Earth
The shape international mileage calculator of the Earth point to point mileage calculator more closely resembles a flattened spheroid with extreme values for the radius of curvature, or us mileage calculator arcradius, of 6335.437 km at the equator (vertically) and 6399.592 km at air mileage calculator the poles, and having an average greatcircle radius of 6372.795 km.
Using a calculator driving mileage sphere with a radius of 6372.795 km thus results in an error of calculator fuel mileage up to about 0.5%.
A worked example
In order to use this formula for anything practical you canadian fuel mileage calculator will need two sets of coordinates. For example, the latitude and longitude of gas mileage car calculator two airports:
 Nashville International Airport (BNA) in Nashville, TN, USA: N 36°7.2', W 86°40.2
 Los Angeles mileage calculator between cities International Airport (LAX) in Los Angeles, CA, USA: N 33°56.4', W 118°24.0'
You will skagit county mileage calculator first have to convert these coordinates to whole degrees and us highway mileage calculator radians before you can use them effectively in a formula. After conversion, the coordinates become:
Using these values california travel maps, driving mileage calculator in the angular distance equation:

Thus the distance between LAX convert fuel mileage calculator and BNA is about 2887 km or 1794 miles.
Spherical coordinates
In the spherical coordinates used distance mileage calculator for uk by mathematicians and physicists, usually when considering other spheres flight mileage calculator than the Earth's surface, the greatcircle distance is found as free online mileage calculator follows. If is the azimuthal angle and the colatitude, then the spherical distance is given by

Note that if highway mileage trip calculator then and the distance interoffice mileage calculator formula reduces to
The notation above is that used in the physical sciences. mileage between cities calculator Pure mathematicians, by contrast, conventionally interchange the roles of the letters and .
See also
 Flight planning
 Spherical geometry
 Spherical trigonometry
 Central angle
 Haversine formula
 Geodesy
External links
 GreatCircle at MathWorld
 U. S. Census Bureau Geographic Information Systems mileage calculator for ontario canada FAQ, What is the best way to calculate the distance between 2 points? mileage calculator for usa (broken link; content has been mirrored here)
 Great Circle Distance mileage calculator from city to city Airline Mileage Calculator A tool for computing frequent flyer miles using great circle mileage distance calculator distances.
 Aviation Formulary
 Great Circle Distance Calculator Distances generated by site users showing maps and travel times
 Haversine formula in online fuel mileage calculator JavaScript Haversine and other formulae for calculating distances, bearings, 995 fuel for palm os palm gas mileage calculator etc
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