| In figure, a square of diagonal 8 cm is inscribed in a circle, then the area of the shaded region is |
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A) \[\left( 32\pi -16 \right)\,c{{m}^{2}}\]
B) \[\left( 16\pi -32 \right)\,c{{m}^{2}}\]
C) \[\left( 11\pi -32 \right)\,c{{m}^{2}}\]
D) \[\left( 16\pi -64 \right)\,c{{m}^{2}}\]
Correct Answer: B
Solution :
Let the side of a square be a and the radius of circle be r. Given that, length of diagonal of square \[=8\,cm\] \[\Rightarrow \,\,a\sqrt{2}=8\] \[\Rightarrow \,\,a=4\sqrt{2}\,cm\] Now, Diagonal of a square = Diameter of a circle \[\Rightarrow \]Diameter of circle = 8 \[\Rightarrow \]Radius of circle \[=r=\frac{Diameter}{2}\] \[\Rightarrow \,\,r=\frac{8}{2}=4\,\,cm\] \[\therefore \] Area of circle \[=\pi {{r}^{2}}=\pi {{\left( 4 \right)}^{2}}=16\pi c{{m}^{2}}\] and Area of square \[={{a}^{2}}={{\left( 4\sqrt{2} \right)}^{2}}=32c{{m}^{2}}\] So, the area of the shaded region = Area of circle - Area of square \[=\left( 16\pi -32 \right)\,c{{m}^{2}}\] Hence, the required area of the shaded region is \[\left( 16\pi -32 \right)\,c{{m}^{2}}\].
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