if the difference between an Interior angle of a regular polygon of N sides and an exterior angle of a regular polygon of N + 1 sides is equal to 95 degree find the value of n
Answers
The sum of exterior angles of any polygon (not regular only) is always 360°.
So,
=> Sum of exterior angles of (N + 1) sided regular polygon = 360°
=> One exterior angle of (N + 1) sided regular polygon = 360°/(N + 1)
As the same,
=> Sum of exterior angles of N sided regular polygon = 360°
=> One exterior angle of N sided regular polygon = 360°/N
And we know that the sum of one interior angle and one exterior angle of a regular polygon is 180°.
So,
=> One interior angle of N sided regular polygon = 180° - (360° / N)
And now we keep in mind.
As no. of sides of a regular polygon increases, the interior angle also increases, thereby decreasing the exterior angle, so,
Interior angle of N sided regular polygon > Exterior angle of (N + 1) sided regular polygon
Now, given that the difference between the interior angle of N sided regular polygon and exterior angle of (N + 1) sided regular polygon is 95°.
So,
180° - (360° / N) - (360°/(N + 1)) = 95°
=> (360° / N) + (360° / (N + 1)) = 180° - 95°
=> 360°[1/N + 1/(N + 1)] = 85°
=> 1/N + 1/(N + 1) = 85° / 360°
=> (N + 1 + N) / (N(N + 1)) = 17 / 72
=> (2N + 1) / (N² + N) = 17 / 72
=> 72(2N + 1) = 17(N² + N)
=> 144N + 72 = 17N² + 17N
=> 17N² + 17N - 144N - 72 = 0
=> 17N² - 127N - 72 = 0
=> 17N² - 136N + 9N - 72 = 0
=> 17N(N - 8) + 9(N - 8) = 0
=> (N - 8)(17N + 9) = 0
=> N = 8 ; N = - 9 / 17
But since N is a natural number, it can't be - 9 / 17, hence the answer is,
N = 8
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