Six [congruent](https://artofproblemsolving.com/wiki/index.php/Congruent) [circles](https://artofproblemsolving.com/wiki/index.php/Circle) form a ring with each circle [externally tangent](https://artofproblemsolving.com/wiki/index.php/Externally_tangent) to two circles adjacent to it. All circles are [internally tangent](https://artofproblemsolving.com/wiki/index.php/Internally_tangent) to a circle $C$ with [radius](https://artofproblemsolving.com/wiki/index.php/Radius) 30. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$ (the [floor function](https://artofproblemsolving.com/wiki/index.php/Floor_function)).
[2005 AIME I Problem 1.png](https://artofproblemsolving.com/wiki/index.php/File:2005_AIME_I_Problem_1.png)
Define the radii of the six congruent circles as $r$. If we draw all of the radii to the points of external tangency, we get a [regular](https://artofproblemsolving.com/wiki/index.php/Regular_polygon) [hexagon](https://artofproblemsolving.com/wiki/index.php/Hexagon). If we connect the [vertices](https://artofproblemsolving.com/wiki/index.php/Vertex) of the hexagon to the [center](https://artofproblemsolving.com/wiki/index.php/Center) of the circle $C$, we form several [equilateral triangles](https://artofproblemsolving.com/wiki/index.php/Equilateral_triangle). The length of each side of the triangle is $2r$. Notice that the radius of circle $C$ is equal to the length of the side of the triangle plus $r$. Thus, the radius of $C$ has a length of $3r = 30$, and so $r = 10$. $K = 30^2\pi - 6(10^2\pi) = 300\pi$, so $\lfloor 300\pi \rfloor = \boxed{942}$.