The area of a sector a circle with diameter 12cm is 66cm2. If the sector is folded to form a cone, calculate…

Analyze the question: A sector is cut from a circle of diameter 12 cm, so the sector's radius is R = 6 cm (this becomes the slant height of the cone). The sector's area is 66 cm². We need the base radius r of the cone.

Concept: When a sector is folded into a cone:

• Slant height l = radius of the sector = 6 cm
• Arc length of the sector = circumference of the cone's base = 2πr
• Curved surface area of the cone = area of the sector = πrl

Solution:

Area of the sector = πrl

66 = (22/7) × r × 6

66 = (132/7) r

r = 66 × 7 / 132 = 462/132 = 3.5 cm

Check using the sector angle: Area = (θ/360) × π × 6² → 66 = (θ/360) × (22/7) × 36 θ = 66 × 360 × 7 / (22 × 36) = 210° Arc length = (210/360) × 2 × (22/7) × 6 = 22 cm 2πr = 22 → r = 22 / (2 × 22/7) = 3.5 cm ✓

(Note: the sector's area of 66 cm² is less than the full circle's area of about 113 cm², so it is a valid sector.)

Answer: The base radius is 3.5 cm, which is option B.

Tip: For cone-from-sector problems, remember that the sector's area equals the cone's curved surface area, πrl, and the sector's radius equals the slant height l. You can often solve directly for r without finding the angle.