How can you use Pick's Theorem to find the area of a circle?

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Oddman Profile
Oddman answered
Pick's theorem gives a method of calculating the area of a polygon drawn on a uniform grid. It seems to require the vertices of the polygon be grid points. It can be used to approximate the area of a circle to the extent that the circle can be approximated by a polygon whose vertices are grid points.

So, to use Pick's theorem to find the area of a circle, you must make a polygon that is an approximation of the circle. That polygon must have its vertices on grid points. The area of the polygon will be the number of interior grid points plus half the number of boundary grid points less one. The area will be in "square grid units". If the polygon is a good approximation of the circle of interest, then its area can be taken to be the approximate area of the circle.
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Oddman commented
What you ask is apparently not possible using that theorem. It has a very specific domain of application. Circles are not included.
Oddman commented
Think a bit about what you mean by "an exact answer". If the circle dimensions are exact (rational numbers), the area will be irrational, due to the fact that pi is irrational. You cannot get an accurate value for the area by counting dots.

If the area is a rational number, the dimensions of the circle involve the square root of pi, a value that cannot be known exactly. (It is possible to construct such a circle, but it is impossible to say what its dimensions are, except approximately.)
Arnon Nakpitugs
Arnon Nakpitugs commented
What other way would you be able to find the remaining area of the spaces between the circle and inscribed polygon?

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