An interior angle of a regular polygon is 5 times each exterior angle. How many sides has the polygon?

Analyze the question: In a regular polygon, each interior angle and its adjacent exterior angle lie on a straight line, so they add up to 180°. We're told the interior angle is 5 times the exterior angle.

Concept:

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

Solution:

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

x + 5x = 180° 6x = 180° x = 30°

Number of sides: n = 360° / exterior angle = 360° / 30° = 12

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

Tip: Whenever a question compares interior and exterior angles, set up "interior + exterior = 180°" first, find the exterior angle, then use n = 360° ÷ exterior angle. If the interior angle is k times the exterior, then n = 2(k + 1). Here, 2(5 + 1) = 12.