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Question
Find the area of shaded region in Fig. 4, where a circle of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm. (Use π = 3.14 and\[\sqrt{3}\] =1.73)
Solution
Given:
Radius of the circle (r) = 6 cm
Side of the equilateral triangle (a) = 12 cm
Now,
Area of the shaded region = Area of the circle with centre O + Area of the equilateral triangle OAB − 2(Area of the sector OLP)
\[= \pi r^2 + \frac{\sqrt{3}}{4} a^2 - 2\frac{\theta}{360^o}\pi r^2 \]
\[ = \pi \left( 6 \right)^2 + \frac{\sqrt{3}}{4} \left( 12 \right)^2 - 2\frac{60^o}{360^o}\pi \left( 6 \right)^2 (\text{Because OAB is an equilateral triangle})\]
\[ = 36\pi + \frac{\sqrt{3}}{4}\left( 144 \right) - 2\frac{60^o}{360^o}\pi\left( 36 \right)\]
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