Distance and course between two positions
Type positions as 32 14.7 N, 32°14′42″N or 32.245, with a minus sign or S and W for south and west; a pair such as 32.245, -66.48167 pasted into either box of a position fills both. The example is Bowditch’s: from 32° 14.7′ N, 66° 28.9′ W to a point near Chesapeake Light.
The great circle is the shortest line over the Earth’s surface, as if there were no land; courses are true, not magnetic. It is not a route by water.
A great circle is the shortest line between two points on the Earth’s surface. From Bowditch’s example position, 32° 14.7′ N, 66° 28.9′ W, to a point near Chesapeake Light, 36° 58.7′ N, 75° 42.2′ W, it is 536.3 nautical miles (617.1 statute miles), starting on a course of 304.5° T. A rhumb line, the straight line on a Mercator chart, holds one course, 302.0° T, and is 536.5 nautical miles, 0.2 nautical miles longer. On this passage the two hardly differ; the great circle saves more on long passages that run east or west in high latitudes. Type any two positions above for both.
Great circle distance (Bowditch): cos D = sin L1 × sin L2 + cos L1 × cos L2 × cos DLo · Nautical miles = D in degrees × 60 · Initial course: tan C = sin DLo ÷ (cos L1 × tan L2 − sin L1 × cos DLo)
How the Great Circle Distance Is Calculated
L1 and L2 are the latitudes of the two points (south negative) and DLo the difference of longitude between them, the short way round. The formula gives the arc between the points in degrees; each degree is 60 nautical miles, because one minute of arc of a great circle is one nautical mile on the navigator’s sphere. This is the method of great circle sailing by computation in The American Practical Navigator (Bowditch), the U.S. government’s navigation manual. The calculator uses the haversine form of the same formula, which stays accurate for short distances, and finds the quadrant of the course from the signs of the two parts.
Bowditch’s second example runs from 28° N, 122° W to 24° S, 151° E: the arc is 98.55°, or 5,913.2 nautical miles, and the initial course is 247.3° T, as in the manual, which prints 5,913.1 nautical miles by computation and 5,913.2 from the tables of Pub. No. 229. In its first example, from 22° S, 116° E to 20° S, 31° E, Bowditch rounds the arc to 78.23° and prints 4,693.8 nautical miles; unrounded, the arc is 78.2256° and the distance 4,693.5 nautical miles, as the tables of Pub. No. 229 give it and the calculator shows, with an initial course of 253.0° T.
Great Circle or Rhumb Line?
A great circle is the intersection of the Earth’s surface with a plane through its center, and its arc is the shortest distance between two points. On a Mercator chart it bends toward the nearer pole, and its direction changes continually, so navigators pick waypoints along it, often every 5° of longitude, and steer rhumb lines between them. A rhumb line crosses every meridian at the same angle: it keeps one true course and plots as a straight line on a Mercator chart, but it is longer. The saving of the great circle grows with latitude, the farther a track runs east or west and the longer it is (Bowditch).
| Example (Bowditch) | Great circle | Rhumb line | Rhumb line longer by |
|---|---|---|---|
| A ship at 32° 14.7′ N, 66° 28.9′ W to a point near Chesapeake Light (art. 1219) | 536.3 NM, 304.5° T | 536.5 NM, 302.0° T | 0.2 NM (under 0.1%) |
| Leaving Cape Town, 33° 53.3′ S, 18° 23.1′ E, for a point near Ambrose Light, 40° 27.1′ N, 73° 49.4′ W (art. 1220) | 6,762.7 NM, 304.5° T | 6,788.9 NM, 311.1° T | 26.2 NM (0.4%) |
| 28° N, 122° W to 24° S, 151° E (art. 1208, example 2) | 5,913.2 NM, 247.3° T | 5,922.5 NM, 238.2° T | 9.3 NM (0.2%) |
| 22° S, 116° E to 20° S, 31° E (art. 1208, example 1) | 4,693.5 NM, 253.0° T | 4,762.5 NM, 271.4° T | 68.9 NM (1.5%) |
The rhumb lines are worked out by Mercator sailing on the same sphere as the great circle, so the two tracks compare like with like: along the equator or a meridian they come out equal, as they must, since such a track is itself a great circle (Bowditch). Bowditch’s worked answers use the meridional parts of its Table 6, made for the WGS 84 ellipsoid, and come out slightly different: 301.8° T and 538.9 NM for the first example, with the course rounded to N 58.2° W, and 310.9° T and 6,811.5 NM for the second. Along a parallel, the rhumb line is longer by a larger share the higher the latitude. For 50° of longitude due west:
| Latitude | Great circle | Rhumb line (due west) | Rhumb line longer by |
|---|---|---|---|
| 0° (equator) | 3,000.0 NM | 3,000.0 NM | 0.0 NM (0.0%) |
| 30° N | 2,576.3 NM | 2,598.1 NM | 21.8 NM (0.8%) |
| 50° N | 1,891.5 NM | 1,928.4 NM | 36.8 NM (1.9%) |
| 60° N | 1,463.9 NM | 1,500.0 NM | 36.1 NM (2.5%) |
A Straight Line, Not a Route by Water
The calculator measures over the Earth’s surface as if there were no land. A real passage has to go around coasts, islands and shoals: Bowditch notes that a great circle route can’t cross land and shouldn’t take a vessel into dangerous waters. Coastal trips are rarely a single straight line. To measure a computed route over water between two ports or places, use our Sea Distance Calculator; for passage planning, plot the route on a current nautical chart.
Entering Positions
Each box takes one latitude or one longitude: decimal degrees (32.245), degrees and decimal minutes (32 14.7 N) or degrees, minutes and seconds (32 14 42 N), with N, S, E or W, or a minus sign for south and west. A pair copied from a map, such as 32.245, -66.48167, can be pasted into either box of a position and fills both. To change a position from one format to another, use the Latitude and Longitude Converter. The calculator doesn’t look up places; a chart, a cruising guide or your GPS gives the coordinates.
How Accurate Is It?
The calculator treats the Earth as a sphere with 60 nautical miles to a degree, as Bowditch’s great circle formulas do. On the real, slightly flattened Earth a degree of latitude is 59.7 nautical miles long at the equator and 60.3 at the poles (Bowditch, Table 7), so the calculator’s distances can be off by up to about half a percent from a survey-grade calculation on the WGS 84 ellipsoid. Courses are true; to steer by a magnetic compass, apply the variation printed on your chart and your compass’s deviation. To convert a distance, use the Nautical Miles to Kilometers Converter; our guide How to Read a Nautical Chart shows how to measure distance on a chart’s latitude scale.
Frequently Asked Questions
What is a great circle distance?
The shortest distance between two points on the Earth’s surface, along the circle whose plane passes through the Earth’s center. Measured in degrees of arc, each degree is 60 nautical miles.
How do you calculate great circle distance?
Use cos D = sin L1 × sin L2 + cos L1 × cos L2 × cos DLo, where L1 and L2 are the latitudes and DLo the difference of longitude, then multiply D in degrees by 60 for nautical miles. For example, 28° N, 122° W to 24° S, 151° E is 98.55°, or about 5,913 nautical miles.
What is the difference between a great circle and a rhumb line?
A great circle is the shortest route, but its course changes all the way. A rhumb line keeps one course and is a straight line on a Mercator chart, but it is longer, by more the farther the route runs east or west at high latitude.
Why does the course change along a great circle?
The great circle crosses each meridian at a different angle, so to follow it you change course as you go. Navigators pick waypoints along it and steer a constant course between them.
Is the great circle distance the distance by boat?
Only in open water. It ignores land and shoals, so a passage along a coast or between islands is longer. Measure a route over water with a route calculator and check it on a nautical chart.
Is the initial course true or magnetic?
True. To steer by a magnetic compass, apply the variation shown on your chart and your compass’s deviation.
Sources
- NGA, The American Practical Navigator (Bowditch), Vol. I (2019), chapter 12, The Sailings: arts. 1202 (great circles), 1204 (rhumb lines), 1207–1208 (great circle sailing by Pub. No. 229 and by computation, and their examples), 1218–1220 (parallel and Mercator sailing and their examples), read Oct 21, 2026.
- NGA, Bowditch, Vol. II (2019): Table 6, meridional parts for the WGS 84 ellipsoid and their formula, whose first term, for the sphere, the calculator uses; Table 7, length of a degree of latitude; read Oct 21, 2026.
- NIST, Special Publication 811, Appendix B.8: 1 nautical mile = 1,852 meters and 1 mile = 1,609.344 meters, exactly (read Oct 21, 2026).