AMC8 2026

AMC8 2026 · Q11

AMC8 2026 · Q11. It mainly tests Circle theorems, Area & perimeter.

Squares of side length $1, 1, 2, 3,$ and $5$ are arranged to form the rectangle shown below. A curve is drawn by inscribing a quarter circle in each square and joining the quarter circles in order, from shortest to longest. What is the length of the curve?

边长分别为 $1, 1, 2, 3$ 和 $5$ 的正方形排列成下图所示的长方形。在每个正方形内都内切一个四分之一圆，并按从最短边到最长边的顺序将这些四分之一圆连接成一条曲线。该曲线的长度是多少？

(A) \ 4\pi \ 4\pi

(B) \ 6\pi \ 6\pi

(D) \ 8\pi \ 8\pi

(E) \ 13\pi \ 13\pi

Answer

Correct choice: (B)

正确答案：（B）

Solution

We notice that we have multiple quarter-arcs (i.e. $\frac{1}{4}$ of a full circumference) of varying radii. However, since they all have the same formula, $\frac{1}{4} \cdot 2\pi r = \frac{\pi}{2}r$, we can take out $\frac{\pi}{2}$ by Distributive Property. Thus we have \[\frac{\pi}{2} (1 + 1 + 2 + 3 + 5) = \frac{\pi}{2} (12) = \boxed{ \textbf{(B) } 6\pi }\] as our total curve length.

我们注意到有多个不同半径的四分之一圆弧（即整个圆周的 $\frac{1}{4}$）。但是由于它们都有相同的长度公式，即 $\frac{1}{4} \cdot 2\pi r = \frac{\pi}{2} r$，我们可以运用分配律将 $\frac{\pi}{2}$ 提出来。因此曲线长度为 \[ \frac{\pi}{2} (1 + 1 + 2 + 3 + 5) = \frac{\pi}{2} \times 12 = \boxed{\textbf{(B) } 6\pi} \] 。

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