A sector of 120 degrees cut out from a circle has an area of 66/7 sq cm (the text may appear as 667 sq cm, meaning 66/7). Find the radius of the circle (in cm), assuming pi = 22/7.

Introduction / Context:This question tests the area-of-sector concept. A sector is a fraction of the full circle determined by its central angle. The sector area formula is: Sector area = (theta/360) * pi * r^2. Here theta is 120 degrees, so the sector is exactly one-third of the circle. That means the full circle’s area is 3 times the sector area. Once we convert sector area into full circle area, we can use the standard circle area formula A = pi*r^2 to solve for r. The problem is designed to work neatly with pi = 22/7 and an area written as 66/7 sq cm (which can sometimes be mistyped as 667).

Given Data / Assumptions:

• Sector angle theta = 120 degrees
• Sector area = 66/7 sq cm
• pi = 22/7
• Sector area formula: (theta/360)*pi*r^2

Concept / Approach:Since 120/360 = 1/3, we have (1/3)*pi*r^2 = 66/7. Multiply both sides by 3 to get pi*r^2, then solve for r using pi = 22/7.

Step-by-Step Solution: (120/360) = 1/3 (1/3) * pi * r^2 = 66/7 pi * r^2 = 3 * (66/7) = 198/7 Substitute pi = 22/7: (22/7) * r^2 = 198/7 Cancel /7: 22*r^2 = 198 r^2 = 198/22 = 9 r = 3 cm

Verification / Alternative check:Full circle area with r=3 is (22/7)*9 = 198/7. One-third of that (for 120 degrees) is (198/7)/3 = 66/7, matching the given sector area exactly.

Why Other Options Are Wrong: 1 cm and 2 cm produce much smaller areas because area grows with r^2. 4 cm and 6 cm produce much larger sector areas than 66/7 for 120 degrees.

Common Pitfalls:Forgetting the (theta/360) factor, using 120/180 instead of 120/360, or not squaring r properly when rearranging the equation.

Final Answer:The radius of the circle is 3 cm.