We want to find the area of the red D-shaped area ADBC below:

 

We find the area of the sector AOBC, then the area of 
triangle AOB, and then subtract to find the area of the 
red area, the segment ADBC.

We find the area of the sector AOBC.

Since AB = 15 in., AD = 15/2 = 7.5 in.,
Given OD = 5 in.

So tan(∠AOD) = 7.5/5 = 1.5
∠AOD in radians = 0.9827937232
∠AOB = 2(0.9827937232) = 1.965587446
We find the (radius)² = OA² by the Pythagorean theorem:

OA² = OD²+AD²
OA² = 5²+7.5²
OA² = 81.25 = (radius)²

Area of sector AOBC = (radius)²(q)

Area of sector AOBC = (81.25)(1.965587446) = 79.85199001 sq. in. 

Area of triangle AOB = (base)(height) = (AB)(OD)

Area of triangle AOB = (15)(5) = 37.5 sq.in.

Finally we subtract (Area of sector AOBC) - (Area of triangle AOB)

Answer:  79.85199001 sq. in. - 37.5 sq.in. = 42.35199001 sq. in.

Edwin