The ratio of the areas of two regular octagons which are respectively inscribed and circumscribed to a circle of radius $r$ is

cos \frac{π}{8}

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{sin}^{2} \frac{π}{8}

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{cos}^{2} \frac{π}{8}

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{tan}^{2} \frac{π}{8}

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Solution

The correct option is C ${cos}^{2} \frac{π}{8}$
Inscribed circle of a regular polygon of $n$ sides
$A_{1} = n r^{2} tan \frac{π}{n}$
Here $n = 8$
$∴ A_{1} = 8 r^{2} tan \frac{π}{8}$
Circumscribed circle of a regular polygon of $n$ sides is
$A_{2} = \frac{n R^{2}}{2} sin \frac{2 π}{n}$
For $n = 8$ we have
$A_{2} = \frac{8 R^{2}}{2} sin \frac{2 π}{8}$
$= \frac{8 R^{2}}{2} sin \frac{π}{4}$
$= \frac{8 r^{2}}{2} 2 sin \frac{π}{8} cos \frac{π}{8}$ (for $R = r$ )
$= 8 r^{2} sin \frac{π}{8} cos \frac{π}{8}$
$∴ \frac{A_{2}}{A_{1}} = \frac{8 r^{2} sin \frac{π}{8} cos \frac{π}{8}}{8 r^{2} tan \frac{π}{8}}$
$= \frac{sin \frac{π}{8} cos \frac{π}{8}}{\frac{sin \frac{π}{8}}{cos \frac{π}{8}}}$
$= {cos}^{2} \frac{π}{8}$

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