see the attached picture to better understand the problem
we know that the area of the shaded portion in the circle=area 1+area 2
step 1 find area 1 area 1=area of semi circle area 1=pi*r²/2 diameter=12 units radius r=12/2----> 6 units
r=6 units area 1=pi*6²/2----> 56.52 units²
step 2 find the area 2
area 2=area sector CADC-area triangle ACD
in the right triangle ABC BC=3 AC=r-----> AC=6 AB=? applying Pythagoras Theorem AC²=AB²+BC²------> AB²=6²-3²-----> AB²=27----> AB=3√3 units area triangle ACD=b*h/2 b=2*3√3---> 6√3 units h=3 units area triangle ACD=6√3*3/2----> 9√3 units²-----> 15.59 units²
find the angle ACB tan ∠ACB=AB/BC-----> 3√3/3---> √3 ∠ACB=arc tan (√3)-----> 60°
the central angle ACD=60*2-----> 120°
area of sector CADC=(120/360)*pi*r²----> (120/360)*pi*6²---> 37.68 units² area 2=area sector CADC-area triangle ACD area 2=37.68-15.59----> 22.09 units²
step 3 the area of the shaded portion in the circle=area 1+area 2 the area of the shaded portion in the circle= 56.52 +22.09----> 78.61 units²
the answer is the area of the shaded portion is 78.61 units²
