Find the area of the shaded portion in the circle.
2 answers:
See the attached picture to better understand the problem
we know that
<span>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*r</span>²/2
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
we know that
<span>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*r</span>²/2
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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