On a red background, a white spotted ball contains thirty (spherical cap shaped) spots. The circumference of the ball's great circle is $54 \mathrm{~cm}$, and the circumference of the spots is $11 \mathrm{~cm}$. What percentage of the ball's surface area is covered by the spots?
Let the radius of the sphere corresponding to the ball be $R$, and its center be $O$. The radius of the base circle of the spherical cap corresponding to the dot be $\varrho$, its center be $K$, and the height of the spherical cap be $m$.

The radius of the sphere and the base circle of the spherical cap can be directly calculated from the given data:

$$
R=\frac{54}{2 \pi} \approx 8.59 ; \quad \varrho=\frac{11}{2 \pi} \approx 1.75
$$

Consider a dot (spherical cap) on the sphere. The two opposite points of the base circle of the spherical cap on the sphere are $A$ and $B$.

Cut the sphere with a plane passing through $A$, $B$, and the center of the sphere $O$. This plane intersects the sphere in a great circle. The intersection of the line $O K$ and the sphere is point $C$.

![](https://cdn.mathpix.com/cropped/2024_05_02_11f2011e444524aacd35g-1.jpg?height=461&width=425&top_left_y=564&top_left_x=839)

$O C=O B=R, K B=\varrho, K C=m, O K=R-m$. For the right triangle $O K B$, apply the Pythagorean theorem: $R^{2}-\varrho^{2}=(R-m)^{2}$, from which:

$$
R-m=\sqrt{R^{2}-\varrho^{2}} \approx 8.41
$$

From this, $m=0.18$.

Calculate the surface area of the sphere and the spherical cap. The surface area of the sphere is: $4 \pi R^{2}=4 \pi \cdot 8.59^{2}$. The surface area of the spherical cap is: $2 \pi R m = 2 \pi \cdot 8.59 \cdot 0.18 \approx 9.72$. The desired ratio is:

$$
\frac{30 \cdot 9.72}{927.25} \cdot 100 \approx 31.45 \%
$$