The ratio of the exterior angle to the interior angle of a regular polygon is 1:11. How many sides has the polygon?

Analyze the question: We have a regular polygon where exterior : interior = 1 : 11. We need the number of sides, n.

Concept: At each vertex, the interior and exterior angles lie on a straight line, so:

• Interior angle + Exterior angle = 180°
• Each exterior angle of a regular polygon = 360°/n

Solution:

The total number of parts in the ratio is 1 + 11 = 12.

Exterior angle = (1/12) × 180° = 15°

Now use the exterior angle formula:

360°/n = 15°

n = 360°/15° = 24

Check: Interior angle = 180° − 15° = 165°. Ratio 15 : 165 = 1 : 11 ✓

Answer: The polygon has 24 sides, which is option B.

Tip: For any ratio problem on exterior and interior angles, always add the ratio parts and divide 180° (since they are supplementary). Then use n = 360°/exterior angle. Quick shortcut: if the ratio is 1 : k, then n = 360/(180/(k+1)) = 2(k+1). Here, 2(12) = 24.