JEE Main & Advanced Mathematics Trigonometric Equations Regular Polygon

Regular Polygon

Category : JEE Main & Advanced

A regular polygon is a polygon which has all its sides equal and all its angles equal.

(1) Each interior angle of a regular polygon of n sides is $\left( \frac{2n-4}{n} \right)\times$right angles $=\left[ \frac{2n-4}{n} \right]\times \frac{\pi }{2}$ radians.

(2) The circle passing through all the vertices of a regular polygon is called its circumscribed circle.

If $a$ is the length of each side of a regular polygon of $n$ sides, then the radius $R$ of the circumscribed circle, is given by  $R=\frac{a}{2}\,.\,\text{cosec }\left( \frac{\pi }{n} \right)$

(3) The circle which can be inscribed within the regular polygon so as to touch all its sides is called its inscribed circle.

Again if $a$ is the length of each side of a regular polygon of $n$ sides, then the radius $r$ of the inscribed circle is given by $r=\frac{a}{2}\,.\,\text{cot }\left( \frac{\pi }{n} \right)$

(4) The area of a regular polygon is given by $\Delta =n\,\,\times$ area of triangle $OAB$

$=\frac{1}{4}n{{a}^{2}}\cot \,\left( \frac{\pi }{n} \right),$    (in terms of side)

$=n{{r}^{2}}\,.\,\tan \,\left( \frac{\pi }{n} \right),$      (in terms of in-radius)

$=\frac{n}{2}\,.\,{{R}^{2}}\sin \,\left( \frac{2\pi }{n} \right),$ (in terms of circum-radius)

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