Spherical law of cosines distance calculator

Enter both coordinate pairs in decimal degrees to receive the central angle and distance in kilometers, miles, and meters. It is a compact alternative to the haversine formula and is especially convenient when a direct cosine expression suits an existing calculation, spreadsheet, lesson, or implementation. Results use a fixed mean Earth radius, making every request deterministic and easy to reproduce.

Enter and interpret the coordinate pairs

Provide the latitude and longitude of the starting point as lat1 and lng1, then provide the destination as lat2 and lng2. Values must be decimal degrees: latitude runs from -90 at the South Pole to 90 at the North Pole, while longitude runs from -180 to 180. Negative latitude identifies the Southern Hemisphere, and negative longitude conventionally identifies positions west of Greenwich. The calculator returns the central angle in degrees together with kilometers, miles, and meters. The distance is the shorter great-circle arc over the surface of the model sphere, not a straight line through Earth. Reversing the points produces the same distance. Identical coordinates produce zero, subject only to the limits of floating-point arithmetic. Check coordinate order carefully because swapping a latitude with a longitude can still create syntactically valid numbers while identifying a completely different location. Keep enough decimal places for your intended scale: fewer decimals describe a less precise position even though the formula itself remains deterministic.

How the spherical law of cosines works

The method converts both latitudes and the longitude difference from degrees to radians. It then computes the cosine of the central angle using sin(latitude one) times sin(latitude two), plus cos(latitude one) times cos(latitude two) times cos(longitude difference). Applying arccosine gives the angular separation at Earth’s center. Multiplying that angle in radians by the fixed mean radius of 6,371 kilometers gives the surface distance. The same result is converted to meters and statute miles for convenience. Floating-point calculations can occasionally produce a cosine infinitesimally above 1 or below -1, especially for coincident or nearly antipodal points. The implementation clamps that intermediate value to the valid arccosine interval, preventing an otherwise valid calculation from becoming not-a-number. Output rounding is stable for repeatable API fixtures and comparisons. Unlike an ellipsoidal geodesic solver, this formula deliberately treats Earth as a sphere, so it favors clarity and compactness over survey-grade modeling of Earth’s flattening.

Choose this method for the right accuracy needs

Use spherical law-of-cosines distance for education, quick geographic comparisons, map summaries, routing estimates, dataset checks, and applications where a spherical approximation is appropriate. It is easy to audit because the calculation is a single familiar trigonometric identity. For most ordinary point separations, it gives a practical great-circle estimate comparable in purpose to the haversine method. At extremely short separations, the arccosine step can lose numerical precision because the cosine is very close to one; haversine is traditionally preferred when tiny distances must remain well resolved. For surveying, legal boundaries, high-precision navigation, or long baselines where Earth’s ellipsoidal shape matters, use an ellipsoidal method such as Vincenty or a modern geodesic library instead. Also remember that this result ignores terrain, road networks, buildings, restricted areas, currents, and flight procedures. It measures idealized surface separation, not a trip itinerary. Record the radius assumption when comparing this output with another tool, since two spherical calculators using different Earth radii can legitimately return slightly different distances.