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Explanation:
The radius of the larger circle is CB = CA+AB = 8+3 = 11 inches.
The area of the larger circle with this radius is...
A = pi*r^2
A = pi*11^2
A = 121pi
The area of the smaller circle (radius CA = 8 inches) is
A = pi*r^2
A = pi*8^2
A = 64pi
The difference is 121pi-64pi = 57pi
This represents the exact area of the full ring. Refer to figure 1 below. Specifically the red region of that figure.
However, we don't want the full ring. We only want a small fraction of it (as figure 2 shows, which is the original diagram your teacher gave you). Specifically, we want (4*30)/360 = 120/360 = 1/3 of the full ring. The 4*30 refers to the idea we have four copies of the 30 degree angle.
So we'll take 1/3 of the 57pi to get (1/3)*57pi = (57/3)pi = 19pi
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The value 19pi is the exact area of the entire shaded region.
From here, you can use a calculator to determine that
19pi = 19*3.1415926535898 = 59.6902604182061 which rounds to 59.7
If your teacher insists you use pi = 3.14, then 19*pi = 19*3.14 = 59.66 which also rounds to 59.7
