The interior angles of a polygon with n sides, are in an A.P. with common difference 6º. If the largest interior angle of the polygon is 219º, then n is equal to __________. [2025]

Question

The interior angles of a polygon with n sides, are in an A.P. with common difference $6^{o}$ . If the largest interior angle of the polygon is $219^{o}$ , then n is equal to __________. [2025]

Smart Achievers · Accepted Answer

(20)

Sum of interior angles $= (n - 2) \times 180^{o}$

$\Rightarrow \frac{n}{2} (2 a + (n - 1) 6) = (n - 2) \times 180$

$\Rightarrow a n + 3 n^{2} - 3 n = (n - 2) \times 180$ ... (i)

Now, according to question a + (n – 1)6 = 219

$\Rightarrow a = 225 - 6 n$ ... (ii)

Putting value of $a$ from equation (ii) to (i), we get

$225 n - 6 n^{2} + 3 n^{2} - 3 n = 180 n - 360$

$\Rightarrow 3 n^{2} - 42 n - 360 = 0$

$\Rightarrow n^{2} - 14 n - 120 = 0$

$\Rightarrow n = 20, - 6$ (Rejected)

Solution

Q.
The interior angles of a polygon with n sides, are in an A.P. with common difference $6^{o}$ . If the largest interior angle of the polygon is $219^{o}$ , then n is equal to __________. [2025]

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Solution

Q. The interior angles of a polygon with n sides, are in an A.P. with common difference 6o. If the largest interior angle of the polygon is 219o, then n is equal to __________. [2025]

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Q.
The interior angles of a polygon with n sides, are in an A.P. with common difference $6^{o}$ . If the largest interior angle of the polygon is $219^{o}$ , then n is equal to __________. [2025]