The area of the shaded region in the given circle is [tex]\rm 120 +36\sqrt{3} \ m^2\\[/tex].
We have to determine
What is the area of the shaded region in the given circle in terms of pi and in simplest form?
According to the question
The radius of the circle r is 12 m.
And the sector angle is 60 degrees.
In the given figure the area of the shaded region is found by adding the area of the circle and the area of the triangle and subtracting the area of the sector.
What is area of circle?
The area of a circle is defined as the number of square units that cover inside the circle.
[tex]\rm Area \ of \ circle = \pi r\\
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Area \ of \ circle = \pi (12)^2\\
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Area \ of \ circle = \pi \times 144\\\\ Area \ of \ circle = 144\pi [/tex]
A whole of a circle surrounds 360°, thus the ratio of the sector's angle calculation to 360° is directly proportional to the fraction of the circle's area is computed.
[tex]\rm Area \ of \ sector = \dfrac{Secotr \ angle }{360} \times \pi r^2\\
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Area \ of \ sector =\dfrac{60}{360} \times 144\pi \\
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Area \ of \ sector= \dfrac{1}{6} \times 144\pi \\
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Area \ of \ sector =24\pi [/tex]
The area of the triangle is half of the product of the base into height.
[tex]\rm Area \ of \ triangle = \dfrac{1}{2} \times Base \times height\\
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Area \ of \ triangle = \dfrac{1}{2} \times (12 cos30) \times (12)\\
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Area \ of \ triangle = \dfrac{1}{2} \times 12\times \dfrac{\sqrt{3}}{2} \times 12\\
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Area \ of \ triangle = \dfrac{1}{2} \times= 36\sqrt{3}[/tex]
Therefore,
The area of the shaded region = Area of the circle - Area of the sector + Area of the triangle.
The area of the shaded region = [tex]\rm 144\pi - 24\pi + 36\sqrt{3}\\
\\
\\[/tex]
The area of the shaded region = [tex]120 +36\sqrt{3} \ m^2\\[/tex]
Hence, the area of the shaded region in the given circle is [tex]\rm 120 +36\sqrt{3} \ m^2\\[/tex].
To know more about the Area of the circle click the link given below.
https://brainly.com/question/1238286
