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Find the area of the shaded region in the figure

What steps should I do? I tried following the steps listed here https://answers.yahoo.com/question/index?qid=20100305030526AAef8nZ

But I got 150.7 which is wrong.

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  • $\begingroup$ Use the same idea of my answer here. math.stackexchange.com/questions/402858/… $\endgroup$ – Luis Felipe Jul 23 '15 at 19:31
  • $\begingroup$ Find area of sector. Find area of triangle. Subtract area of sector from area of triangle. Now subtract that from $144\pi$. $\endgroup$ – Zain Patel Jul 23 '15 at 19:42
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Hint: Break the shaded region up into two shapes. One is a portion of the circle (you know the portion because of the given angle), and the other is a triangle (which is equilateral). Find the area of each shape and then add them.

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The basic idea is right. First calculate the area of the sector by observing that $\dfrac{ \text{area of sector} }{ \text{area of circle} } = \dfrac{\pi / 3}{2 \pi}$.

Then find the area of the triangle by noting that the triangle is equilateral, and subtract it off. This will give you the white region.

Now subtract the white region from the area of the circle.

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  • $\begingroup$ I think you mixed up your last line. $\endgroup$ – hexaflexagonal Jul 23 '15 at 19:44
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    $\begingroup$ @DivergentQueries Sure did. Edited, and thanks. $\endgroup$ – Ken Jul 23 '15 at 19:45
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Area of white strip=Area of sector subtending$\frac{\pi}{3}$ at the center-Area of equilateral triangle

Area of sector subtending$\frac{\pi}{3}$ at the center=$\frac{1}{2}\theta r^2=\frac{1}{2}\frac{\pi}{3} (12)^2=24\pi=24\times 3.14=75.36 $sq units

Area of equilateral triangle=$\frac{\sqrt3}{4}$(side)$^2$=$\frac{\sqrt3}{4}$(12)$^2$=36$\sqrt3=$62.35 sq units

Area of white strip=75.36 -62.35=13.01 sq units

Area of circle=$\pi r^2=3.14\times12\times12=452.16$sq units

Shaded area=Area of circle-Area of white strip=452.16-13.01=439.15 sq units

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Notice, the area of the equilateral triangle (indicated by red) $$=\frac{1}{2}(12)(12)\sin\frac{\pi}{3}$$ $$=72\frac{\sqrt{3}}{2}= 36\sqrt{3}$$ The area of the major circular sector subtending at the center an angle $2\pi-\frac{\pi}{3}=\frac{5\pi}{3}$ $$=\frac{1}{2}\left(\frac{5\pi}{3}\right)(12)^2=120\pi$$ Hence, the area of the shaded portion $$=\text{area of (red) triangle}+\text{area of major sector}$$ $$=\color{blue}{36\sqrt{3}+120\pi\approx 439.34494 \ }$$

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