Beck’s Theorem: Calculate the Area – Without a Difficult Formula

Thus it turns out that the formation of two polygons that follow Beck’s theorem also satisfies the theorem automatically. This can be demonstrated in a similar way: if you join two polygons s₁ And the s₂ to greater swhere this and one of the smallest (s₁) satisfies Beck’s theorem, then does the rest s₂. Knowing this, one can now prove that the theorem is valid for all triangles that fulfill the requirements.

Step 3: The triangles satisfy Beck’s theorem

To do this, you start with a right-angled triangle whose legs are aligned horizontally and vertically. Now you can invert this in its chord and get a rectangle that satisfies Pick’s theorem as already shown: A = I + B/ 2 – 1. Once again one can use that I = 2I + J – 2 where I’ are the interior points of triangles. In addition to: B = 2 (with me + 1) where B’ are the edge points of the triangles. Entering it into the zone one gets: a = 2I + J – 2+ with me + 1 – 1 = 2 (I’ + B’/ 2-1). there a Twice the area of ​​the triangle a’ Corresponding, one can see that both triangles are subject to Beck’s theorem.

Now one just has to show that the other triangles also follow the theorem. It’s easy to do: to do this, you draw the smallest rectangle around a given triangle. Since all right triangles follow Beck’s theorem and all rectangles as well, one can use the rule of composition to show that the triangle under study is also subject to the theorem (since it applies to all other shapes within the rectangle).

Step Four: From Triangles to General Polygons

You’re done with that, because connected polygons that don’t intersect can always be divided into triangles. And because these all satisfy Beck’s theorem, so does the aggregate construct. This is a great handy way to calculate the area of ​​complex polygons in no time at all.

But often one finds oneself faced with the measurement of more complex numbers. For this, Pick’s theorem can also be useful – because many of them can at least be approximated by polygons. In addition, many scientists have now worked on extensions of the useful theory, for example to take into account shapes with holes or objects in higher spatial dimensions. As it turns out, engineering is sometimes easier than you think.

What is your favorite math theory? Feel free to write it in the comments – and maybe it will be the topic of this column soon!