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Q.
$H_1$ is a regular hexagon circumscribing a circle. $H_2$ is a regular hexagon inscribed in the circle. Find the ratio of areas of $H _1$ and $H _2$.
Mensuration
Solution:
Let $H_1$ be $P Q R S T U$ and $H_2$ be $A B C D E F$.
Let $R$ be the radius of the circle.
$\therefore R=\frac{\sqrt{3}(P U)}{2} \Rightarrow P U=\frac{2 R}{\sqrt{3}}$
$A B=R(\because$ A hexagon inscribed in a circle must have its side equal to the radius of the circle).
Area of $H_1=\frac{3 \sqrt{3}}{2}\left(\frac{2}{\sqrt{3}} R\right)^2$
And area of $H_2=\frac{3 \sqrt{3}}{2}(R)^2$
$\left(\because \text { Area of hexagon }=\frac{3 \sqrt{3}}{2} \times(\text { Its side })^2\right)$
$\text { Required ratio }=\frac{\left(\frac{2}{\sqrt{3}} R\right)^2}{R^2}=\frac{4}{3}$
