The interior angle of a regular polygon is 6 times its exterior angle find the number of sides of the polygon.

Analyze the question: We have a regular polygon whose interior angle is 6 times its exterior angle. We need the number of sides, n.

Concept:

• Interior angle + exterior angle = 180° (they lie on a straight line)
• Each exterior angle of a regular polygon = 360°/n

Solution:

Let the exterior angle be x. Then the interior angle is 6x.

x + 6x = 180° 7x = 180° x = 180°/7 ≈ 25.71°

Number of sides:

n = 360° / x = 360 × 7 / 180 = 14

Check: Exterior angle = 360/14 = 180/7°. Interior angle = 180 − 180/7 = 1080/7°, and 6 × 180/7 = 1080/7°. ✓

Tip: For any "interior is k times exterior" question, the exterior angle is 180°/(k+1), so n = 360 ÷ (180/(k+1)) = 2(k+1). Here k = 6, so n = 2(7) = 14. This shortcut saves time in the exam.