Question : A square and a regular hexagon are drawn such that all the vertices of the square and the hexagon are on a circle of radius $r$ cm. The ratio of area between the square and the hexagon is:
Option 1: $3: 4$
Option 2: $4:3\sqrt{3}$
Option 3: $\sqrt{2}:\sqrt{3}$
Option 4: $1:\sqrt{2}$
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Correct Answer: $4:3\sqrt{3}$
Solution :
Let the side of a hexagon be $a$.
Area of a hexagon = $\frac{3\sqrt{3}}{2}×a^2$
Let the radius of the circle be $r$.
Diagonal of a square = $2r$
Area of a square $=\frac{(\text{Diagonal})^2}{2}=\frac{(2r)^2}{2}=2r^2$
Since the side of a regular hexagon inscribed in a circle is equal to the radius of the circle, $a=r$
So, the area of a hexagon = $\frac{3\sqrt{3}}{2}×r^2$
Area of a square : Area of a hexagon
= $2r^2:\frac{3\sqrt{3}}{2}×r^2$
= $4:3\sqrt{3}$
Hence, the correct answer is $4:3\sqrt{3}$.
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