Perimeter of regular hexagon inscribed in a circle: If a regular hexagon is inscribed in a circle of radius r, what is its perimeter?

Introduction / Context:
In a regular hexagon inscribed in a circle, each side equals the circle’s radius. This geometric fact allows a fast perimeter computation without trigonometry.

Given Data / Assumptions:

• Regular hexagon with 6 equal sides
• Inscribed in a circle (circumradius = r)

Concept / Approach:
Each central angle is 360/6 = 60 degrees, making each chord equal to the radius. Therefore, side length s = r and perimeter P = 6s = 6r.

Step-by-Step Solution:
s = rP = 6 * s = 6r

Verification / Alternative check:
Construct equilateral triangles by joining the center to adjacent vertices; each has side r, confirming the chord length equals r.

Why Other Options Are Wrong:
3r, 9r, and 12r are incorrect multiples; 6πr confuses polygon perimeter with a circle’s circumference formula.

Common Pitfalls:
Assuming perimeter equals circumference; forgetting the chord property for regular hexagons.

Final Answer:
6r