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

We are given that the interior angles are in arithmetic progression (A.P.) 
with a common difference of 6° and 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 first angle. Simplifying: \[ an + 3n^2 - 3n = (n-2) \times 180 \] Now, using the condition that the largest interior angle is 219°, we have: \[ a + (n-1) \times 6 = 219 \] which simplifies to: \[ a = 225 - 6n \] Substitute this value of \( a \) into the sum equation: \[ (225 - 6n) + 3n^2 - 3n = (n-2) \times 180 \] Solving the resulting quadratic equation gives \( n = 20 \).