Question

The interior angles of a polygon with \( n \) sides, are in an A.P. with common difference 6°. If the largest interior angle of the polygon is 219°, then \( n \) is equal to:

• A. 20
• B. 18
• C. 25
• D. 15

Hint:

When dealing with arithmetic progressions in geometry, use the standard formulas for sum and difference of angles to set up and solve equations.

Answer 1

The Correct Option is A

The interior angles of an n-sided polygon form an arithmetic progression (A.P.) with a common difference of 6°. The largest angle is 219°. The sum of the interior angles of an \( n \)-sided polygon is calculated as: \[ \frac{n}{2} \left( 2a + (n-1) \times 6 \right) = (n-2) \times 180 \] where \( a \) represents the first angle. Simplifying this equation yields: \[ an + 3n^2 - 3n = (n-2) \times 180 \] Given that the largest interior angle is 219°, we can establish the following relationship: \[ a + (n-1) \times 6 = 219 \] This simplifies to: \[ a = 225 - 6n \] Substituting this expression for \( a \) into the sum equation results in: \[ (225 - 6n) + 3n^2 - 3n = (n-2) \times 180 \] Solving the subsequent quadratic equation for \( n \) yields \( n = 20 \).