AMC 8 2026 Problem 11

AMC8 2026 年第 11 题

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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 的正方形被排列成如下所示的矩形。在每个正方形内画一个四分之一圆，并按从小到大的顺序将这些四分之一圆连接起来形成一条曲线。这条曲线的长度是多少？

A. $4\pi$
B. $6\pi$
C. $\displaystyle (\frac{13}{2})\pi$
D. $8\pi$
E. $13\pi$

正确答案 · Correct Answer

解析 · Solution

解法1

各段均为四分之一圆弧，弧长

\displaystyle = \frac{\pi}{2}r

。提取公因子

\displaystyle \frac{\pi}{2}

：总长

\displaystyle = \frac{\pi}{2} \times (1+1+2+3+5) = \frac{\pi}{2} \times 12 = 6\pi

。

Solution 1

Each arc is a quarter-circle with arc length

\displaystyle \frac{\pi}{2}r

. Extracting the common factor

\displaystyle \frac{\pi}{2}

: total

\displaystyle = \frac{\pi}{2} \times (1 + 1 + 2 + 3 + 5) = \frac{\pi}{2} \times 12 = 6\pi

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