Find the area of the shaded portion in the circle.

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

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