The interior angle of a regular polygon is five times the size of its exterior angle. Identify the polygon.

Analyze the question

We have a regular polygon whose interior angle is 5 times its exterior angle. We need to find the number of sides and name the polygon.

Concept

For any polygon, the interior and exterior angles at a vertex lie on a straight line, so:

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

Solution

Let the exterior angle = x. Then the interior angle = 5x.

x + 5x = 180° 6x = 180° x = 30°

Number of sides:

n = 360° / 30° = 12

A 12-sided polygon is a dodecagon.

Check: The interior angle = 5 × 30° = 150°. Using the formula (n − 2) × 180°/n = 10 × 180°/12 = 150°. ✓

Checking the other options

• Enneadecagon = 19 sides (exterior angle ≈ 18.9°, not a match)
• Icosagon = 20 sides (exterior angle 18°, interior 162°, which is 9 times, not 5)
• Hendecagon = 11 sides (exterior angle ≈ 32.7°, not a match)

Exam tip: When a question gives a ratio between the interior and exterior angles, always use "interior + exterior = 180°". Then find n from 360° ÷ exterior angle. Also remember the common names: hendecagon = 11, dodecagon = 12, icosagon = 20.