Prove As shown in Fig. 7-8, a circle with any grid point as the center and a radius of 1 contains only one integer point inside, and the area of the circle is $\pi$. If the area of $\odot P$ is greater than $\pi$, then the radius of $\odot P$ is greater than 1, and the center should fall within or on the edge of a square with a side length of 1, denoted as $A B C D$. If $\odot P$ contains only 1 grid point inside, then at most one of $A, B, C, D$ is inside this circle, thus at least one pair of vertices is not inside $\odot P$, let's assume they are $A$ and $C$. Since the radius of $\odot P$ is greater than 1, and $A$ and $C$ are not inside $\odot P$, the center $P$ is outside the circle $\odot A$ with $A$ as the center and a radius of 1, and also outside the circle $\odot C$ with $C$ as the center and a radius of 1. However, $\odot A$ and $\odot C$ already cover the square $A B C D$.
Therefore, $O$ is not inside or on the boundary of $A B C D$, which contradicts the assumption.
In conclusion, the maximum area of a circle containing only one grid point inside is $\pi$.
Example 1 Find the maximum area of a circle that contains only one lattice point inside.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 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