Christmas Tree Farm

Geometry Level 5

A Christmas tree farmer decides to plant pine trees up in hilly property that he owns. Measuring at a constant shortest ground distance from the center, he marks off a circular perimeter that is $$2000\pi$$ meters long. He plants the pine trees at an uniform density of one tree per $$10$$ square meters. When completed, and after the entire area inside the $$2000\pi$$ meter long circular perimeter is planted with pine trees, he notices that he still has a lot of surplus unplanted pine trees, which means he used less than his original calculated estimates. He then decides to mark off the shortest lines between $$4$$ points equally spaced along the $$2000\pi$$ meter circular perimeter, and then counts all the trees. He discovers that instead of the expected ratio of $$2:\pi$$ between the tree count inside the $$4$$ sided boundary and the total tree count inside the $$2000\pi$$ meter circular perimeter, he has the ratio of $$1:2$$. He scratches his head and wonders what's going on.

What is the area of the property inside the $$2000\pi$$ meter circular perimeter?

If $$A$$ is the area in square meters, find $$\left\lfloor \dfrac { A }{ 1000 } \right\rfloor$$

Note: The Gaussian curvature everywhere on the property inside the $$2000\pi$$ meter circular perimeter is $$0$$, with the allowed exception of a single point. (This means that any patch of the property, which does not include that single point, can be "laid flat", i.e. every point on the surface is locally like a cylinder, where $$K=0$$. This is not true for spheres, where $$K > 0$$, or saddles, where $$K < 0$$.)

There are no lakes, sinkholes, etc.

Only elementary math is needed to solve this one, no calculus nor differential geometry is needed here.

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