This is area addition and subtraction

The area of the shaded region is 818.43

Step-by-step explanation:

First, we need to calculate the area of the gray part (look at the first picture) . For do that we use a rule of three taking into account the following:

The total circle have a measure in grades of 360°, and for know the total area we use the equation:

where is 3.1416 and r is the circle's ratio . So the total area is:

Area=2427.94

then, the gray part of the circle (look first at the picture)  have a measure in grades of 150°.

So the rule of three  is like:  if 2427.94 is the area of the entire circle or 360°, then the area of 150° of the circle is:

2427.94 ----------> 360°

    X ---------------------------------------> 150°

Finding X, we obtain:

X=1011.64

So the gray part of the circle have 1011.64

After found the area of the gray part, we will continue subtracting the area of the triangle.

For do that we need to know that the area of the triangle is given by the expression:

Triangle Area = B*H/2

Where B is the base´s length of the triangle  and H is the altitude´s length of the triangle . At the second picture we can see what is B (base) and what we will name the H (altitude).

Therefore:

B = 27.8 in

Now, For know the altitude we must use a trigonometric property, the sine:

Sine theta = O / Hip

Where O is the opposite side to the angle and Hip is the triangle´s largest side and theta is the known angle.

The angle that we will use is 30° (look at the second attachment), therefore:

Sin 30° = H/B

Where B is the base´s length of the triangle  and H is the altitude´s length of the triangle .

So, solve for H, our unknown variable, we obtain:

H = B*sin 30°

H = 27.8 in *1/2

H = 13.9 in

Therefore the area of the triangle is:

Triangle Area = B*H/2  

Triangle Area =27.8 in * 13.9 in/2

Triangle Area =193.21

Finally, The shaded area is the gray area less the triangle area:

SA = GA - TA

Where SA is Shaded area , GA is gray area  and TA is Triangle area

Replacing:

SA = 1011.64 - 193.21

SA = 818.43

At the end the area of the shaded region is 818.43

the answer has got to be 20