The area of a 120° sector of a circle is 66/7 cm²; find the diameter of the…

Concept: The area of a sector of a circle with central angle θ and radius r is given by Area = (θ/360°) × πr². This scales the full circle's area (πr²) by the fraction of 360° that the sector's angle occupies.

Application: substitute the given values and solve for the radius, then the diameter.

1. Given: θ = 120°, sector area = 66/7 cm², π = 22/7.

2. Substitute into the sector-area formula: (120/360) × (22/7) × r² = 66/7.

3. Simplify the angle fraction: 120/360 = 1/3, so (1/3) × (22/7) × r² = 66/7, i.e. (22/21) × r² = 66/7.

4. Solve for r²: r² = (66/7) × (21/22) = (66 × 21)/(7 × 22) = 9.

5. Take the square root: r = 3 cm.

6. Convert radius to diameter: diameter = 2r = 2 × 3 = 6 cm.

Cross-check: substituting r = 3 cm back into the sector-area formula gives (1/3) × (22/7) × 3² = (1/3) × (22/7) × 9 = 198/21 = 66/7 cm², which reproduces the given sector area and confirms the radius.

So the diameter of the circle is 6 cm.