Determining the area of ​​a complex figure using probability theory

Why determine the area of ​​a complex figure?

Who knows why? For example, it became necessary to determine the area of ​​the territory on the map. Of course, you can look in the directory or search the Internet, but sometimes the territories are non-standard - for example, you attend to the problems of forests in the Amazon floodplain and want to measure the area of ​​green spots on satellite photos every month. If you are a nerd (in a good sense of the word), then you may need to measure the leaf area of ​​different varieties of the same plant. Or, for example, a more prosaic task - you need to putty a piece of the wall, and the cans of putty are only enough for 1 square meter. m. - you need to find out whether to buy one can or fork over two.

What is the difficulty of finding the area?

Of course, if the shape is a rectangle, a circle or, worse, an ellipse, then the problem is solved with the help of Google and a calculator. But where would you find a formula, or simpler, for finding the area, say, of such a pattern?

Probability theory, your way out!

At once I will make a reservation that the theory of probability in its essence does not imply the exact solution of problems. So it will be in this case - if you need cosmic accuracy, then I suggest digging in the direction of simulation methods. If the error within 2-5% suits you perfectly, then the same calculator, basic programming skills, and the ability to count to one hundred will suffice.
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The essence of the method

The essence of the method is simple to banality. Suppose we put a cabbage leaf (on a remark about the biologist above) on a cloudy day on a rectangular pan, and put the pan under a drizzling rain. And then we spotted a certain time (for example, five minutes) and calculated how many droplets fell on the tray, and how many directly on the sheet. If we take into account that rain usually drips evenly, it turns out to be a simple proportion - a sheet is so many times smaller than a pallet, how many times less raindrops fell on it than on the whole pallet.

Back to our figure

So, how to determine the area of ​​the pink peverny? Yes, it is very simple - to enclose the figure in rectangular borders and set randomly many points. The more, the better (in accordance with the law of large numbers). And then count the number of points that fell on the figure.

I deliberately do not discuss the implementation of such an algorithm, because there are a lot of options. You can just close your eyes and randomly poke a ballpoint pen, and you can act more scientifically - using programming languages. For example, the PHP code took no more than 15 lines from me, and the result was this:


The points totaling 300 are, of course, affixed using a random number generator. For convenience of counting points, I divided the image into 36 sectors - now I need to count the number of points that fell on the image in each sector, and add the results. We will summarize the data in the table (the table cell corresponds to the sector in the picture):

0	four	eight	four	0	0
0	7	five	6	0	four
3	6	13	7	eight	five
one	ten	ten	13	7	2
0	2	3	7	ten	2
0	0	2	five	3	0

Now we have all the data to calculate the pink spot area:
the area of ​​the rectangle described is 20 cm x 20 cm = 400 square meters. cm;
the number of points in the rectangle is 300;
the number of points inside the figure (the sum of the values ​​from the table) - 157;
the area of ​​the figure is 209.33 square meters. cm.

And how accurate is that?

Indeed, it remains to determine the accuracy of this method. Of course, everything depends on the number of points, and here it is necessary to observe the golden mean - ten for our example would be clearly not enough, and from a thousand would be too much in the eyes. Therefore, we will try to determine the error for three hundred points and the described square with a side of 20 cm. To do this, take a figure whose area is known to us in advance. For example, such:


Dotting points:


The results are recorded in the table:

0	6	eleven	eight	five	0
9	15	eight	five	13	2
eleven	eight	five	14	13	five
ten	eleven	eight	eight	four	four
2	14	9	ten	four	one
0	3	five	6	0	0

Calculate the area of ​​the figure:
the area of ​​the rectangle described is 20 cm x 20 cm = 400 square meters. cm;
the number of points in the rectangle is 300;
the number of points inside the figure (the sum of the values ​​from the table) - 237;
the area of ​​the figure is 316 square meters. cm.

It is easy to calculate that the real area of ​​a circle with a radius of 10 cm is 314.16 square meters. see. Thus, the error of the method was 0.59%, which in most cases is sufficient for applied use.