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As stated in the title, I wonder how large the area covered by certain coordinates actually is, when they have the accuracy XX°YY'ZZ", for example:

51°26'02"N, 07°20'04"E

How can this be calculated?

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    $\begingroup$ A second of arc is about 31 meters of latitude, but, as @ebv notes, the longitude length depends on latitude and has a length of cosine of the latitude times about 31 meters. However, there's a difference between precision and accuracy. For example, you can give a city's latitude/longitude to within an arcsecond precision, but you'd just be giving it for a specific point in the city, since most cities are larger than an arcsecond squared. $\endgroup$ – user967 Dec 9 '19 at 18:07
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This is a coordinate, a single point. No area or distance can be calculated from that, sorry ...

https://en.wikipedia.org/wiki/Geographic_coordinate_system

The length of one arcsecond depends on the latitude. The Wikipedia article explains it quite well. Distances between two coordinates can be calculated via the great circle distance. Calculating areas on earth's surface is more involved, since they require a projection into 2D.

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As others have said, a single set of coordinates gives a point rather than an area.

However, I assume that the intent of the question is an understanding of the precision of these coordinates - i.e. what's the area covered by the 1"x1" shape comprising all of the lat and lon values that would round to a whole number of seconds. The answer to that is "it depends".

The distance covered by one arc minute of latitude, by definition, is one nautical mile. One nautical mile is 1852 metres, therefore one arc second (1/60 minutes) of latitude represents just under 31m.

So far, so straightfoward. But longitude isn't so simple, because the distance represented by an arc second of longitude is not the same everywhere on the planet - it depends on the latitude. This makes sense if you look at a map and see how the lines of longitude converge - i.e. have less distance between them - towards the poles.

As Barry Carter said in a comment, the distance represented by an arc second of longitude is approximately equal to $31 \cos(lat)$ metres[1]. Hence, the area of a 1" ~square around any given point is equal to $31^2 \cos(lat)$ m2. That will vary from zero[2], at the poles, to 961 m2 on the equator. At the particular coordinates that you gave as an example, it'd be $31 \times 31\cos(51.434^{\circ}) \approx 599$ m2.

[1] Before anybody corrects me, all of the above does assume a spherical earth. In reality It's A Bit More Complicated.

[2] Very nearly zero.

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A point do not have an area.

A point is a 0-dimensional mathematical object which can be specified in n-dimensional space using an n-tuple (x1, x2, ..., xn) consisting of n coordinates.

The area of a point cannot be calculated. So the area must be null.

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