The region shown below consists of 24 squares, each with side length 1 centimeter. What is the area, in square centimeters, of the largest circle that can fit inside the region, possibly touching the boundaries?

A. $3\pi$

B. $4\pi$

C. $5\pi$

D. $6\pi$

E. $8\pi$

1. 2025 AMC 8 官方解析

Answer (C):

Let point $O$ be located at the center of the region. The largest circle centered at $O$ will intersect the boundary of the region at eight points, as shown in the figure above. The radius $r$ of the circle will correspond to the hypotenuse of a right triangle with leg lengths of 2 and 1 cm. By the Pythagorean Theorem, $r = \sqrt{2^2 + 1^2} = \sqrt{5}$ cm. No other circle inside the region will be larger because any other center will be closer to at least one of eight boundary points. Therefore the area of the largest circle that can fit inside the region is $\pi r^2 = 5\pi\text{ cm}^2$.