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.

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.