A 120-degree sector cut from a circle has area 66/7 square centimetres. Find the radius of the circle.

Introduction / Context:Sector area links a portion of a circle to its radius through the central angle fraction. By equating the given sector area to the fractional part of the full circle area, the radius follows directly.

Given Data / Assumptions:

• Sector angle = 120 degrees.
• Sector area = 66/7 cm^2.
• Full circle area = pi * r^2.

Concept / Approach:The sector with central angle theta has area (theta/360) * pi * r^2. With theta = 120, this factor is 120/360 = 1/3. So (1/3) * pi * r^2 = 66/7. Solve for r using a convenient rational value for pi = 22/7 to keep arithmetic clean.

Step-by-Step Solution:

Verification / Alternative check:Back-substitute: Full area = pi * r^2 ≈ (22/7) * 9 = 198/7. Sector is one third of that: (1/3) * 198/7 = 66/7 cm^2, matching the given value exactly.

Why Other Options Are Wrong:

• 1cm or 2cm: Lead to sector areas far below 66/7.
• 4cm: Produces a sector area larger than 66/7 with the given angle.

Common Pitfalls:

• Using 120/180 instead of 120/360 for the sector fraction.
• Mixing centimetres and metres; units must remain consistent.

Final Answer:3cm