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

Step 1: Define the Given Information

The interior angles of a polygon are in 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 given by: \[ \frac{n}{2} \left( 2a + (n-1) \times 6 \right) = (n-2) \times 180 \] where \( a \) is the smallest angle. Simplified equation: \[ an + 3n^2 - 3n = (n-2) \times 180 \]

Step 2: Apply the Condition for the Largest Angle

The largest interior angle is 219°. This yields the equation: \[ a + (n-1) \times 6 = 219 \] Simplified equation: \[ a = 225 - 6n \]

Step 3: Substitute into the Sum Equation

Substitute \( a = 225 - 6n \) into the sum equation: \[ (225 - 6n) + 3n^2 - 3n = (n-2) \times 180 \] Solving this quadratic equation results in: \[ n = 20 \]

Final Answer: \( n = 20 \)