Question

A company is designing a new logo that is comprised of four identical sectors of circles that have central angles

that measure 30º. A portion of the logo is shaded. The distance from the center, C, to point A is 8 inches
and the distance from point A to point B is 3 inches.
Determine the area of the logo that is shaded to the nearest tenth of a
square inch. Show the work that leads to your answer.

Answer 1

Answer: 59.7 square inches

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Step-by-step 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