AMC8 2025

AMC8 2025 · Q12

AMC8 2025 · Q12. It mainly tests Pythagorean theorem, Area & perimeter.

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?

下图所示区域由24个边长1厘米的正方形组成。能放入该区域内最大圆的面积是多少平方厘米，该圆可能触及边界？

(A) \ 3\pi \ 3\pi

(B) \ 4\pi \ 4\pi

(D) \ 6\pi \ 6\pi

(E) \ 8\pi \ 8\pi

Answer

Correct choice: (C)

正确答案：（C）

Solution

The largest circle that can fit inside the figure has its center in the middle of the figure and will be tangent to the figure in $8$ points. By the Pythagorean Theorem, the distance from the center to one of these $8$ points is $\sqrt{2^2 + 1^2} = \sqrt5$, so the area of this circle is π52=(C) 5π.

能放入图形内的最大圆的圆心在图形的中心，将在$8$个点与图形相切。根据勾股定理，从中心到这些$8$个点之一的距离是$\sqrt{2^2 + 1^2} = \sqrt5$，因此该圆的面积是$\pi(\sqrt5)^2=5\pi$，即(C) $5\pi$。

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