下图由 2 个完全相同的大四分之一圆和 2 个完全相同的小四分之一圆组成。大四分之一圆的半径是小四分之一圆半径的两倍。用 $\pi$ 表示该图形的周长。

1. Find the radii of the small and large quarter circles using the total width.

Let the radius of the small quarter circle be $r$ and the large quarter circle be $R$.
Given $R = 2r$.
The total horizontal width is made up of the horizontal radius of the left large quarter circle, the horizontal distance between the centers (which is $r$), and the horizontal radius of the right large quarter circle.
Total width = $R + r + R = 2R + r$
Substitute $R = 2r$:
$2(2r) + r = 30$
$5r = 30 \implies r = 6\text{ cm}$
$R = 2(6) = 12\text{ cm}$

2. Calculate the total length of the straight line segments of the perimeter.

The straight edges of the figure consist of:

• The top-left horizontal edge of the large quarter circle: $R$
• The bottom-right horizontal edge of the large quarter circle: $R$
• Two vertical segments connecting the large and small quarter circles, each of length $R - r$.
  Total straight length = $R + R + 2(R - r) = 4R - 2r$
  Substitute $R = 12$ and $r = 6$:
  Total straight length = $4(12) - 2(6) = 48 - 12 = 36\text{ cm}$

3. Calculate the total length of the curved arcs.

The curved parts consist of 2 large quarter circle arcs and 2 small quarter circle arcs.
Length of 2 large quarter circle arcs (1 large semi-circle) = $\frac{1}{2} \times 2\pi R = \pi R = 12\pi\text{ cm}$
Length of 2 small quarter circle arcs (1 small semi-circle) = $\frac{1}{2} \times 2\pi r = \pi r = 6\pi\text{ cm}$
Total arc length = $12\pi + 6\pi = 18\pi\text{ cm}$

4. Find the total perimeter.

Total perimeter = Total straight length + Total arc length
Total perimeter = $(36 + 18\pi)\text{ cm}$