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# Geometry/area between concentric polygons

Question
QUESTION: There are 2 concentric polygons (one inside the other).  The outer one is always 100 meters from the boundary of the inner one.  As the inner one increases in area from 0.01 to 100 hectares, at what point does the area in the zone between the two polygons exceed the area of the inner polygon? And is there a formula that would apply to all problems like this?

ANSWER: The area of a polygon with n sides and radius r measured to the center of a side is given by
n*r²*tan(180°/n).  To find when the area of the outside polygon is twice the area of the inside polgon (making the area of the interior be the same as the area between them), find the value of this formula at r and 100+r.

Doing this gives us 2*n*r²*tan(180°/n) = n*(r+100)²*tan(180°/n).
The n*tan() term can be divided out, giving 2r² = r²+200r+10,000.
This means solving the parabola r² - 200r - 10000 = 0.

Use the quadratic equation to find r, the radius of the inner polyugon.

---------- FOLLOW-UP ----------

QUESTION: Just one clarification: My polygon is a circle.  So do I just leave out n (the number of sides) or should I use a differently formulated equation?

It is not important how many sides the polygon has, it is always true.
Given it has n sides, then it can also have n+1 sides.
Given the last line, n would be allowed to increase indefinitely.
Eventually, n will get so large that it appears to be a circle.

Geometry

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