# Area--it's more than length x width

April 8, 2008 7:30 AM Subscribe

How do geographers calculate the area of a landmass, given crinkly coastlines? Or to put it another way, if I have a map of a large island X, how do I calculate its area in square kilometres?

When I've had to get a fast 'n' nasty area of some kind of irregular shape, I've found it relatively simple to just do a sort of longhand calculus (don't run away yet!) solution for it. Basically, I'll divide the area into bars of a certain constant width, then measure the approximate length of all the bars, add them, and then multiply by the width of the bars. The narrower the bars, the more accurate the measurement, but the more measurements you have to make.

So, say you're trying to measure the square kilometrage of Manhattan. You could, for example, start at the south end and draw a series of parallel lines across the width of the island 1 kilometer apart until you get to the north end, so that the island looks like it's got a bunch of stripes across it. Measure the width of the island at every line, and then add up all the measurements. Since the interval used is only 1 kilometer (meaning you'd only be multiplying by 1 anyway), the sum of all the measurements would roughly equal the area of the island.

I have no idea how actual geographers would go about it, but if you're looking for a fairly close approximation, this isn't a bad method.

posted by LionIndex at 7:42 AM on April 8, 2008 [1 favorite]

So, say you're trying to measure the square kilometrage of Manhattan. You could, for example, start at the south end and draw a series of parallel lines across the width of the island 1 kilometer apart until you get to the north end, so that the island looks like it's got a bunch of stripes across it. Measure the width of the island at every line, and then add up all the measurements. Since the interval used is only 1 kilometer (meaning you'd only be multiplying by 1 anyway), the sum of all the measurements would roughly equal the area of the island.

I have no idea how actual geographers would go about it, but if you're looking for a fairly close approximation, this isn't a bad method.

posted by LionIndex at 7:42 AM on April 8, 2008 [1 favorite]

when i had to do this, i used a planimeter.

posted by lester at 7:45 AM on April 8, 2008 [1 favorite]

posted by lester at 7:45 AM on April 8, 2008 [1 favorite]

For what it's worth: the crinkly coastlines don't make so much of a difference in your estimate of area. When geographers try to measure the

posted by wyzewoman at 7:56 AM on April 8, 2008

*length*of a coastline, then the crinkles are critical. Then, essentially, the size of the ruler they use becomes important. If you measure all the one-foot crinkles along the coast of California, you will decide it is much longer than if you only include one-mile crinkles. This is due to the fractal dimension of the coastline.posted by wyzewoman at 7:56 AM on April 8, 2008

The planimeter is the ideal thing. If you don't have one at your disposal, you can make a polygonal approximation of the region to whatever degree of accuracy you want, and then use the so-called surveyor's area formula. It's explained only briefly there, but it's pretty simple, and it computes exactly what the planimeter would compute if you slid it along your straight line approximation.

posted by Wolfdog at 7:58 AM on April 8, 2008

posted by Wolfdog at 7:58 AM on April 8, 2008

Here's a low-tech, less computational but entirely workable solution: you can press silly putty or playdoh out to a uniform thickness, mark the boundary of the figure on it, trim it and throw away the scraps, then re-roll it into a rectangle of the same thickness.

posted by Wolfdog at 8:01 AM on April 8, 2008 [1 favorite]

posted by Wolfdog at 8:01 AM on April 8, 2008 [1 favorite]

A very similar question was asked a few months ago.

In brief, a minimum unit of analysis is defined with respect to the scale of the map (avoiding the problem of fractal-esque coastlines), and spherical coordinates (rather than 2-dimensional map-based referencing systems) are used if the areal extent is sufficiently broad so as to introduce calculation distortion.

posted by zachxman at 8:04 AM on April 8, 2008

In brief, a minimum unit of analysis is defined with respect to the scale of the map (avoiding the problem of fractal-esque coastlines), and spherical coordinates (rather than 2-dimensional map-based referencing systems) are used if the areal extent is sufficiently broad so as to introduce calculation distortion.

posted by zachxman at 8:04 AM on April 8, 2008

I've heard that one approach is to cut out a paper image of it, and then weigh it. (And compare that to the weight of a reference unit square of the same paper.)

posted by Class Goat at 8:09 AM on April 8, 2008

posted by Class Goat at 8:09 AM on April 8, 2008

Wolfdog:

I was going to suggest something similar, or rather, how I would do it. I'd get a map, cover it in rubber cement, let that dry, and then trace the border with string. Then I'd measure the string and divide by 2.

posted by JeremiahBritt at 8:09 AM on April 8, 2008

I was going to suggest something similar, or rather, how I would do it. I'd get a map, cover it in rubber cement, let that dry, and then trace the border with string. Then I'd measure the string and divide by 2.

posted by JeremiahBritt at 8:09 AM on April 8, 2008

Divide the area into squares... count them up. Then go around the coastal squares, ignore the ones that have more sea than land, count the ones that have more land as one, and the ones that seem almost equally divided as 0.5.

posted by fearfulsymmetry at 8:15 AM on April 8, 2008 [1 favorite]

posted by fearfulsymmetry at 8:15 AM on April 8, 2008 [1 favorite]

One can use monte carlo methods to approximate the area using uniform random variables. This can be used to calculate the area of any shape.

Basically, take tons and tons of random points on a 100x100 plot (or any plot), with the shape fully contained in the plot. Count each point in the shape, don't count points not in the shape. Then divide the points in the shape by the total number of points you used to get an area as a percentage of the 100x100 plot. Multiply by a scaling factor to get the actual area.

Can be used to approximate Pi -- see here @ wikipedia

posted by djpyk at 8:26 AM on April 8, 2008

Basically, take tons and tons of random points on a 100x100 plot (or any plot), with the shape fully contained in the plot. Count each point in the shape, don't count points not in the shape. Then divide the points in the shape by the total number of points you used to get an area as a percentage of the 100x100 plot. Multiply by a scaling factor to get the actual area.

Can be used to approximate Pi -- see here @ wikipedia

posted by djpyk at 8:26 AM on April 8, 2008

Jeeze, you people are oldschool.

1- scan the map

2- insert image into AutoCad or other vector drawing program

3- draw a polyline (or equivalent) around the island. Make as precise as you wish.

4- use command "area" on polyline (or 'info', etc.)

posted by signal at 8:26 AM on April 8, 2008

1- scan the map

2- insert image into AutoCad or other vector drawing program

3- draw a polyline (or equivalent) around the island. Make as precise as you wish.

4- use command "area" on polyline (or 'info', etc.)

posted by signal at 8:26 AM on April 8, 2008

This is a really cool problem.

Get a pair of compasses, set them to 1 metre, and work your way around the coast of an island, measuring it. Your compasses will skip over a lot of features that are smaller than a metre - rock pools, little inlets, all the knobbly stuff that makes a coastline look like a coastline.

Now set your compasses to 50cm and start again. This time, some of those little knobbly bits that you skipped over will get caught, and so the length of the coastline will be greater. But you're still skipping features that are less than 50cm in size.

Now set your compasses to 1cm....

So the answer to "how long is a coastline" is "it depends how closely you look".

Check out the Koch Curve for a good model of a coastline. Each time you iterate the curve, it's length increases by 33%, but the area it bounds is finite. So its an infinitely long line folded up into a finite space.

All this is lifted from James Gleick's Chaos, an awesome book. I can't recommend it highly enough.

When I did GCSE maths I got bonus points for estimating the area of the UK using the Monte Carlo method. Most of these approaches boil down to successive approximations of one type or another.

posted by Leon at 8:30 AM on April 8, 2008

Get a pair of compasses, set them to 1 metre, and work your way around the coast of an island, measuring it. Your compasses will skip over a lot of features that are smaller than a metre - rock pools, little inlets, all the knobbly stuff that makes a coastline look like a coastline.

Now set your compasses to 50cm and start again. This time, some of those little knobbly bits that you skipped over will get caught, and so the length of the coastline will be greater. But you're still skipping features that are less than 50cm in size.

Now set your compasses to 1cm....

So the answer to "how long is a coastline" is "it depends how closely you look".

Check out the Koch Curve for a good model of a coastline. Each time you iterate the curve, it's length increases by 33%, but the area it bounds is finite. So its an infinitely long line folded up into a finite space.

All this is lifted from James Gleick's Chaos, an awesome book. I can't recommend it highly enough.

When I did GCSE maths I got bonus points for estimating the area of the UK using the Monte Carlo method. Most of these approaches boil down to successive approximations of one type or another.

posted by Leon at 8:30 AM on April 8, 2008

If you don't have a wide-format scanner and AutoCAD, that's somewhat . . . inconvenient.

Although nowadays, a

If I was just a regular Joe without special equipment or software, I'd probably go for the LionIndex method, especially if the island was small enough to not have to worry about significant distortion due to projection.

posted by that girl at 8:37 AM on April 8, 2008

Although nowadays, a

*geographer*probably would have the map in some electronic format, and be able to manage something approximately equivalent to signal's method.If I was just a regular Joe without special equipment or software, I'd probably go for the LionIndex method, especially if the island was small enough to not have to worry about significant distortion due to projection.

posted by that girl at 8:37 AM on April 8, 2008

Planimeters & Green's Theorem explains the theory of how a planimeter works in a little more detail than the Wikipedia entry linked above.

Areas with Gauss-Green Formula shows how to write a very simple Mathematica program to approximate the area of a shape given a finite number of points on the boundary (you could easily port this to Excel). There are also some exercises that shouldn't require more than a simple graphics program, a spreadsheet and some maps pulled from the internet.

As for the "fractal-esque" coastlines zachxman references, it doesn't really apply to this question. Your area isn't going to diverge just because you measure the perimeter more precisely. The problem only arises when you allow the plane to become a surface embedded in 3-dimensions, so that you can have bumps that contribute arbitrarily to the area, just as the problem of "how long is the perimeter" only arises when you let the curve defining it move through 2-dimensions.

posted by dsword at 8:52 AM on April 8, 2008

Areas with Gauss-Green Formula shows how to write a very simple Mathematica program to approximate the area of a shape given a finite number of points on the boundary (you could easily port this to Excel). There are also some exercises that shouldn't require more than a simple graphics program, a spreadsheet and some maps pulled from the internet.

As for the "fractal-esque" coastlines zachxman references, it doesn't really apply to this question. Your area isn't going to diverge just because you measure the perimeter more precisely. The problem only arises when you allow the plane to become a surface embedded in 3-dimensions, so that you can have bumps that contribute arbitrarily to the area, just as the problem of "how long is the perimeter" only arises when you let the curve defining it move through 2-dimensions.

posted by dsword at 8:52 AM on April 8, 2008

Not ultra-useful, but there's a nice exhibit at the Science Museum in one of the areas that's about 20 years old, with some planimeters in it: I took a photo last time I was there: http://www.flickr.com/photos/adrianhon/373899083/

posted by adrianhon at 9:00 AM on April 8, 2008

posted by adrianhon at 9:00 AM on April 8, 2008

What signal said. I guess the mathematical solution is kind of interesting to figure out, but I'd just dump a scanned image into ArcGIS, georeference it, run a script or something to make drawing the polygon easier and wham, instant area. Tack on a DEM (Digital Elevation Model) for the image and you could get 3 dimensional area too.

I would imagine professional geographers let GIS do all their measuring for them, based on projections from satellite and aerial photos.

posted by elendil71 at 9:02 AM on April 8, 2008

I would imagine professional geographers let GIS do all their measuring for them, based on projections from satellite and aerial photos.

posted by elendil71 at 9:02 AM on April 8, 2008

*If I was just a regular Joe without special equipment or software, I'd probably go for the LionIndex method, especially if the island was small enough to not have to worry about significant distortion due to projection.*

Well, normally, I would actually use the signal method. But I've worked a lot with complex driveways that were built in the fifties and drawn on 36x48 sheets of paper--which means to get a scan, I have to send it out to a reprographics shop. Then to get the area, I have to wait for them to scan it and send me the file, then insert the image into AutoCAD, then scale it appropriately, then spend however long trying to trace it...at that point it's a lot faster to just pull out the parallel ruler and a pencil and use a map that's already perfectly to scale.

posted by LionIndex at 9:45 AM on April 8, 2008

I knew there would be a Google Maps hack for this: Google Planimeter.

Note that the way this works, you need to divide the perimeter up into straight line segments, which you can made as detailed as you want, down to probably a meter or so. (This should eliminate the need to weigh pieces of paper or silly putty, as suggested above.)

See also the very excellent Gmaps Pedometer for calculating trip lengths, shoreline lengths, whatever distance measurements you need.

posted by beagle at 10:49 AM on April 8, 2008

Note that the way this works, you need to divide the perimeter up into straight line segments, which you can made as detailed as you want, down to probably a meter or so. (This should eliminate the need to weigh pieces of paper or silly putty, as suggested above.)

See also the very excellent Gmaps Pedometer for calculating trip lengths, shoreline lengths, whatever distance measurements you need.

posted by beagle at 10:49 AM on April 8, 2008

*if I have a map of a large island X, how do I calculate its area in square kilometres?*

Do you have a map of it in ArcView? Use XTools to find its land area. (speaking practically)

posted by salvia at 11:09 AM on April 8, 2008

As Class Goat says, the old-skool way to integrate irregular shapes was to cut them out and weigh them. You need a good milligram balance and some reference standards (1 square centimeter pieces of paper, or whatever is appropriate). Knowing the scale of your map, you can convert the weight of your island to its area.

This is how we used to integrate chromatography peaks in that strange brief period where strip-chart recorders were available but not computerized. It was a pain in the butt.

posted by Quietgal at 11:37 AM on April 8, 2008

This is how we used to integrate chromatography peaks in that strange brief period where strip-chart recorders were available but not computerized. It was a pain in the butt.

posted by Quietgal at 11:37 AM on April 8, 2008

IAAG, actually, and I think zachxman has it correct.

Practically, though, we just do it in Arc....

posted by troika at 11:28 PM on April 8, 2008

Practically, though, we just do it in Arc....

posted by troika at 11:28 PM on April 8, 2008

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posted by aswego at 7:36 AM on April 8, 2008