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Prove that the area of a regular polygon...

Prove that the area of a regular polygon hawing `2n` sides, inscribed in a circle, is the geometric mean of the areas of the inscribed and circumscribed polygons of `n` sides.

Text Solution

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Let a be the radius of the circle. Then,
`S_(1)` = Area of regular polygon of `n` sides inscribed in the circle
`= (1)/(2) na^(2) sin ((2pi)/(n)) = na^(2) sin.(pi)/(n) cos.(pi)/(n)`
`S_(2) =` Area of regular polygon of n sides circumscribing the circle `= na^(2) tan ((pi)/(n))`
`S_(3) =` Area of regular polygon of 2n sides inscribed in the circle `= na^(2) sin ((pi)/(n))` [replacing n by 2n is `S_(1)`]
`:.` Geometric mean of `S_(1) and S_(2) = sqrt((S_(1) S_(2)))`
`= na^(2) sin ((pi)/(n))`
`= S_(3)`
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