Wikipedia:Reference desk/Archives/Mathematics/2021 September 19

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September 19

Area of center rectangle

For a 2n-polygon with side of 1, what fraction of the area is found in the "center rectangle" formed of the points 1, 2 , n+1 & n+2? It looks somewhat easily to figure out "by hand" for the Hexagon and the Octagon, but beyond that, I'm not sure. It certainly goes to zero and n goes to infinity.Naraht (talk) 06:13, 19 September 2021 (UTC)

2/n catslash (talk) 14:53, 19 September 2021 (UTC)

Let the side length be $s$ and let the distance from the centre of the (regular) polygon to the middle of a side (the apothem) be $h$. Then the area of the polygon can be divided into $2n$ triangles, each of area ${\frac {1}{2}}sh$. The area of the rectangle is $2hs$, so the fractional area is

$f={\frac {2hs}{2n{\frac {1}{2}}sh}}={\frac {2}{n}}$

Sanity check: for a square $f=1$ and for a circle $f=0$. catslash (talk) 15:11, 19 September 2021 (UTC)

Another way to look at this is to divide the rectangle along it's diagonals. Each of the four triangles formed has the same area, so one triangle is 1/4 the area of the rectangle. But, as you pointed out, 2n of the thicker type triangles can be put together to form the entire polygon, and so each triangle is 1/2n the area of the polygon and the result follows. You could express the area of the rectangle and the polygon in terms of sine and cosine, but your way is much more elegant.

Another variation is to divide the rectangle into 8 congruent triangles like the Union Jack. This time 4n of the triangles can be fit together to form the polygon and rest is similar.

If you divide the rectangle along one diagonal, the you get a triangle inscribed in a regular m-gon, with vertices of the triangle also vertices of the polygon, and the area of the triangle is a rational multiple of the area of the m-gon. What other instances of this are there? It seems there are none with m=5, but with m=6 any triangle inscribed in this way works. Are there any instance with odd m>3? --RDBury (talk) 11:58, 20 September 2021 (UTC)

Dissection of the rectangle into 8 triangles, and the polygon into 4n triangles, all congruent, surely wins the prize for elegance. Given a picture, a cat could understand it. catslash (talk) 20:43, 20 September 2021 (UTC)