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The difference between an exterior angle of (n - 1) sided regular polygon and an exterior angle of (n + 2) sided regular polygon is 6°. Find the value of n.

Rectilinear Figures

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Answer

Exterior angle of (n - 1) sided regular polygon = 360°n−1\dfrac{360°}{n - 1}

Exterior angle of (n + 2) sided regular polygon = 360°n+2\dfrac{360°}{n + 2}

Given,

Difference between an exterior angle of (n - 1) sided regular polygon and an exterior angle of (n + 2) sided regular polygon is 6°.

∴360°n−1−360°n+2=6°⇒360°(n+2)−360°(n−1)(n−1)(n+2)=6°⇒360°.n+720°−360°.n+360°n2+2n−n−2=6°⇒1080°n2+n−2=6°⇒1080°=6°(n2+n−2)⇒n2+n−2=1080°6°⇒n2+n−2=180⇒n2+n−2−180=0⇒n2+n−182=0⇒n2+14n−13n−182=0⇒n(n+14)−13(n+14)=0⇒(n−13)(n+14)=0⇒n−13=0 or n+14=0⇒n=13 or n=−14.\therefore \dfrac{360°}{n - 1} - \dfrac{360°}{n + 2} = 6° \\[1em] \Rightarrow \dfrac{360°(n + 2) - 360°(n - 1)}{(n - 1)(n + 2)} = 6° \\[1em] \Rightarrow \dfrac{360°.n + 720° - 360°.n + 360°}{n^2 + 2n - n - 2} = 6° \\[1em] \Rightarrow \dfrac{1080°}{n^2 + n - 2} = 6° \\[1em] \Rightarrow 1080° = 6°(n^2 + n - 2) \\[1em] \Rightarrow n^2 + n - 2 = \dfrac{1080°}{6°} \\[1em] \Rightarrow n^2 + n - 2 = 180 \\[1em] \Rightarrow n^2 + n - 2 - 180 = 0 \\[1em] \Rightarrow n^2 + n - 182 = 0 \\[1em] \Rightarrow n^2 + 14n - 13n - 182 = 0 \\[1em] \Rightarrow n(n + 14) - 13(n + 14) = 0 \\[1em] \Rightarrow (n - 13)(n + 14) = 0 \\[1em] \Rightarrow n - 13 = 0 \text{ or } n + 14 = 0 \\[1em] \Rightarrow n = 13 \text{ or } n = -14.

Since, no. of sides cannot be negative.

∴ n = 13.

Hence, n = 13.

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