Here I am treating only regular polygons , and you will need basic trigonometry. It appears to me the solution for the square barn can actually be universal if we generalize it. In each case the tether describes a sector of circle of its entire radius. Then, at each corner, the next radius is shorter by the length of the side, and the sector is a different percentage. This happens on each side of the polygon. Eventually the tip of the remaining rope from one direction will meet the tip from the other direction and form a triangle, from which you may or may not have to subtract a portion of the barn.
You know all the sides of that triangle and everything else follows from that. In the current case the meeting point is at the center line, so my first instinct was to figure the altitude of the triangle by Pythagoras and work from there. This keeps the math at algebra level without getting into trigonometry.
Since all three sides are known, all three angles and the area can be figured directly from them. This is important because in some case the meeting point is not exactly in the center. The formula for obtaining angles is called the Law of Cosines. There are two area formulas. Heron's Formula requires only the three sides, the other is called SAS because you need to know two sides and the included angle. To keep this page from getting too long, I will give those formulas and others on the Formulas page
This solution is universal because you can use any regular
polygon and any length of rope, and any placement on the polygon.
The length of rope, however, may dictate some differences but still
within the basic procedures. Here's a variation from the internet.
You should have no trouble figuring the area
Below is the triangle from the square barn and three pentagons, showing that wherever you attach the rope, on any polygon, you always wind up with a final triangle. And you always know all three sides so you can figure the area and the angle of the final sector. You can see that the blue dotted line on pentagon 1 shows a case where the final triangle rests entirely on top of the figure.
Now we can see that it is possible to solve these problems with math only. The diagrams help, but all of the necessary information comes with the original statements of type of polygon, length of sides, and length of rope. In the third pentagon you will have to find a different angle for each side.