A circle is inscribed in an equilateral triangle. The circle’s area is 462 sq. cm (use π = 22/7). Find the perimeter of the equilateral triangle.

Introduction / Context:An equilateral triangle has a well-known relation between its side length and the inradius (radius of the inscribed circle). If the inradius is known via the circle’s area, we can back-compute the side length and hence the perimeter of the triangle. The algebra is clean when π = 22/7 and the numbers are crafted to be exact.

Given Data / Assumptions:

• Area of inscribed circle: A_circ = 462 sq. cm.
• π = 22/7.
• For equilateral triangle of side a: inradius r = (a√3) / 6.
• Perimeter P = 3a.

Concept / Approach:From A_circ = πr^2, compute r. Then use r = (a√3)/6 to find a. Finally compute P = 3a. The numbers are chosen so that r comes out as a neat radical, and a, P are integers.

Step-by-Step Solution:r^2 = A_circ / π = 462 * (7/22) = 147r = √147 = 7√3r = (a√3)/6 ⇒ a = 6r/√3 = 6 * (7√3) / √3 = 42Perimeter P = 3a = 3 * 42 = 126 cm

Verification / Alternative check:Compute area from a = 42: inradius r = a√3/6 = 7√3; circle area = πr^2 = (22/7) * 147 = 462 sq. cm, consistent.

Why Other Options Are Wrong:42√3 cms (option A) is the triangle’s side expressed differently (not the perimeter). 72.6 and 168 cms do not match the consistent inradius–side relationships.

Common Pitfalls:Mixing up inradius with circumradius, or using the area formula of the triangle instead of the inscribed circle when working backward; also forgetting that perimeter is 3a.

Final Answer:126 cms