Find the area of the shaded region in the figure
What steps should I do? I tried following the steps listed here
But I got 150.7 which is wrong.
$\endgroup$ 24 Answers
$\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ 2 $\begingroup$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
$\endgroup$ $\begingroup$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 $$=\text{area of (red) triangle}+\text{area of major sector}$$ $$=\color{blue}{36\sqrt{3}+120\pi\approx 439.34494 \ }$$
$\endgroup$
