We are asked to find the area of the minor segment of a circle. The central angle of the segment is $60^{\circ}$ and the radius of the circle is 22 cm.

1. Problem Description

We are asked to find the area of the minor segment of a circle. The central angle of the segment is $60^{\circ}$ and the radius of the circle is 22 cm.

2. Solution Steps

The area of the minor segment is the area of the sector minus the area of the triangle formed by the two radii and the chord.

First, let's find the area of the sector. The area of a sector with central angle $\theta$ (in degrees) and radius $r$ is given by:

$Area_{sector} = \frac{\theta}{360^{\circ}} \pi r^2$

In this case, $\theta = 60^{\circ}$ and $r = 22$ cm.

$Area_{sector} = \frac{60^{\circ}}{360^{\circ}} \pi (22)^2 = \frac{1}{6} \pi (484) = \frac{484\pi}{6} = \frac{242\pi}{3}$ cm$^2$.

Next, let's find the area of the triangle. Since the central angle is $60^{\circ}$ and the two sides are equal to the radius (22 cm), this is an isosceles triangle. Since the angles opposite to equal sides of an isosceles triangle are equal, the other two angles are equal to $\frac{180^{\circ}-60^{\circ}}{2} = \frac{120^{\circ}}{2} = 60^{\circ}$. Thus, this triangle is an equilateral triangle with side length 22 cm.

The area of an equilateral triangle with side length $s$ is given by:

$Area_{triangle} = \frac{\sqrt{3}}{4} s^2$

In this case, $s = 22$ cm.

$Area_{triangle} = \frac{\sqrt{3}}{4} (22)^2 = \frac{\sqrt{3}}{4} (484) = 121\sqrt{3}$ cm$^2$.

Now, we can find the area of the minor segment by subtracting the area of the triangle from the area of the sector:

$Area_{segment} = Area_{sector} - Area_{triangle} = \frac{242\pi}{3} - 121\sqrt{3}$ cm$^2$.

Using $\pi \approx 3.14159$ and $\sqrt{3} \approx 1.73205$,

$Area_{segment} \approx \frac{242 \times 3.14159}{3} - 121 \times 1.73205 \approx \frac{760.26478}{3} - 209.57805 \approx 253.42159 - 209.57805 \approx 43.84354$ cm$^2$.

$Area_{segment} = \frac{242\pi}{3} - 121\sqrt{3} = 121(\frac{2\pi}{3} - \sqrt{3})$

The approximate value is $121(\frac{2 \times 3.14159}{3} - 1.73205) \approx 121(2.09439 - 1.73205) \approx 121(0.36234) \approx 43.84314$ cm$^2$.

3. Final Answer

$\frac{242\pi}{3} - 121\sqrt{3}$ cm$^2$ which is approximately 43.84 cm$^2$.