Question

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

Answer 1

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Answer 2

The value of n is 6.

We know that the sum of the exterior angles of any regular polygon is 360°

So, the exterior angle of n sided regular polygon is = (360/n)°

Similarly, the exterior angle of (n-1) sided regular polygon = 360/(n-1) °

And also the exterior angle of (n+2) sided regular polygon = 360/(n+2) °

It is given that the difference between the (n-1) sided regular polygon and  (n+2) sided regular polygon is 6 °. So, we can represent this statement as:

360/(n-1) 360/(n+2) = 6

On cross multiplying, we have,

360(n+2) - 360(n-1) = 6 (n-1)(n+2)

⇒60(n+2) - 60(n-1) = (n-1)(n+2)                [Dividing both sides by 6]

⇒ 60[(n+2) - (n-1)] = n² + n - 2

⇒ n² +n - 2 = 60[3]

⇒ n² + n - 182 = 0

⇒ n² - 13n + 14n - 182 = 0

⇒ n(n-13) + 14(n-13) = 0

⇒ (n+14)(n-13) = 0

By Zero Product rule, we get, n = -14 and n = 13

Since, side cannot be negative, so n = 13 is accepted.