The subject of mathematics is so serious that it is useful not to miss an opportunity to make it a little entertaining.
(Pascal)
Good afternoon, dear guests and subscribers of my channel!
I remembered a funny incident, how about a year ago I argued with my daughter that I would find the area of any of the presented above polygons in 30 seconds in one action, while she will calculate it with many actions, as taught in school.
Won. The daughter betted ice cream.
And since I remembered this, I want to tell you how easy it is to use one single formula in one action accurately calculate the area of a polygon of any configuration and there is no need to decompose the figure into several the simplest.
But, for such polygons there is one important condition: each vertex must be integer, i.e. to be exactly at the node of the grid.
A mesh is a cell surface on which a figure is depicted.
Node - intersection of grid lines.
The grid can be made with any unit of measurement, because the area is measured in the squares of the selected unit. If the cell is 1x1 cm, then this is 1 sq. Cm, 1x1 m is 1 sq. Cm. etc.
So, there is a very simple formula that connects the area of any polygon with the number of grid nodes located on the boundaries of the shape segments and inside the shape itself. The formula was derived by the Austrian mathematician Georg Alexander Pieck in 1899, after whom it is called by the Pick formula (theorem):
Where:
S is the area of the polygon;
B - the number of nodes inside the figure (pcs.);
Г - the number of nodes located at the vertices and on the segments of the figure (pcs).
To make everything clear, I will give an example with a complex polygon. We need to find the area of the figure below:
Now, we count the nodes located inside, on the vertices and on the segments of the figure. These will be the values of B and G, respectively:
We get that B = 16, G = 7, now it is enough to substitute the values in the formula and we get: S = G / 2 + B - 1 = 7/2 + 16 -1 = 18.5 square units.
Done. The area is 18.5 cells. You can double-check everything and you will be pleasantly surprised!
The pros are that such a formula is easy to remember and easy to use! Of course, there is also a minus, as I mentioned above - the formula does not give an exact result if at least one of the vertices of the polygon is outside the grid node (not integer).
My daughter has already successfully applied this formula in the classroom at school and quickly finds answers, although some teachers disapprove of this approach and still persuade to the classical scheme: divide the polygon into elementary figures, calculate their areas using standard formulas and add them, get result.
But I still think the formula is useful for the speed of calculations. Be sure to tell the kids!
I really hope that you liked the article! Good luck and good!
I offer several publications that will be of interest to you:
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