# Estimating the Distance Between Two Points

I am wondering how to estimate the distance between two points on a map. For example, consider the distance from A to B on the map below. If the latitude and longitude lines shown are separated by 10°, how is it possible to estimate the distance AB.

• but do you have the exact coordinate of those points A and B? Or do you wanna estimate the distance 'approx' directly from the map by hand? – Paul Goyes Mar 1 '20 at 6:06
• @PAULGOYES The latter. I'm very unsure of how this is achieved – JulianAngussmith Mar 1 '20 at 6:46
• I will give you the best option. You can georeference the map and, after that you can measure whatever you want. Do you know how to georeference a map? Is do you like the idea, I can write the short procedure in the answer. But I don't know if this choice do you like. Let me know please. – Paul Goyes Mar 1 '20 at 7:11
• @PAULGOYES I'm not sure what that is. I was reading that it should be possible using some formula of latitude involving cosine. Whatever works, but preferably a method that relies on numerical calculations – JulianAngussmith Mar 1 '20 at 7:28

## 3 Answers

Just to add to other answers:

You can estimate the length of a meridian arc over a spheric surface with the formulas provided by TomazicM.

Also, you can estimate the length of a parallel arc over a spheric surface multiplying the same formula by the cosine of the latitude.

If your points are not along a meridian or a parallel, as JoeBe answer, you can estimate the length of the great circle arc that pass through both points with the Haversine formula.

But let me say, don't calculate the distance. Estimate the coordinates of the points and let GeodSolve to estimate the distance over the ellipsoidal surface. Let me assure you that it is the best algorithm we have available, and that PROJ, QGIS, I imagine that PostGIS also, delegate to this library the calculation of their ellipsoidal distances.

For instance, between points `A = (-30° latitude, 113° longitude)` and `B= (-38° latitude, 111° longitude)` there is a distance (over the WGS 84 ellipsoid) of 906343.630 m. The indeterminacy in the distance is nothing compared to the indeterminacy in the coordinates.

• Just to add more context to the statement "But let me say, don't calculate the distance.": because Earth is not a perfect sphere, the haversine distance or basic distance on a sphere mentioned by @TomazicM will not lead to most accurate results, especially when measuring far distances. The WGS84 elipsoid is the best compromise to match the Earth's form on a global scale and thus may be best for measuring far distances. – JoeBe Mar 4 '20 at 2:53

The "georeferencing" @PAULGOYES is talking about means that you take the image file to a GIS software like QGIS, add a basemap (e.g. OpenStreetMap) and align point on your map to the corresponding points in the basemap. This way you are adding "spatial knowledge" to your image and can measure the distance with a basic distance measure tool in the software. If you want to use a mathematical formula based on estimated guessings of the coordinates in question, you could use the haversine formula

Since question is about estimate, this will be simplified answer on the basic of mathematics as I remember it from high school (45 years ago :-).

If you consider Earth as sphere with radius `R` (of approximately 6,371 km), then circumference is `cf = 2 * Pi * R`. Circle has 360 degrees, and if you have distance of `x` radial degrees, then formula for distance estimation in kilometers would be `d = cf * x / 360`.