The central angle of a sector is 80° and whose length is 96π. What is the radius of the circle?

Correct — B. Arc length = (θ⁄360) × 2πr, so an 80° slice carries 80⁄360 = 2⁄9 of the full circumference.

Arc = (2⁄9) × 2πr = 4πr⁄9
4πr⁄9 = 96π
4r⁄9 = 96 (π cancels from both sides)

r = 96 × 9⁄4 = 216 units → option (b)

Why the others are wrong

• (a)
  196 units — 196 units gives an arc of 4π(196)⁄9 = 784π⁄9 ≈ 87.1π, short of 96π. For 4r⁄9 to land on a whole number the radius has to be a multiple of 9, and 196 is not.
• (c)
  204 units — 204 units gives 816π⁄9 ≈ 90.7π — still under 96π. The gap is small enough to pass for rounding, which is exactly why the value is worth plugging back.
• (d)
  116 units — 116 units gives 464π⁄9 ≈ 51.6π, barely half the stated arc. A circle that small cannot carry a 96π arc across only 80° of turn.

A sector is a fixed fraction of its circle, and the fraction is θ⁄360.

Everything scales by it: the arc is that fraction of the circumference 2πr, the area that fraction of πr². At θ = 80° the fraction is 2⁄9, so the arc is 2⁄9 of the way round.

Because the arc is handed to you as a multiple of π, the π cancels and the equation collapses to one division. Keep π symbolic to the last line — substituting 3.14 early only buys rounding error.

Key facts

• Arc length of a sector = (θ⁄360) × 2πr.
• Area of the same sector = (θ⁄360) × πr², so arc and area share the fraction θ⁄360.
• At θ = 80° that fraction is 80⁄360 = 2⁄9, making the arc 4πr⁄9.
• With r = 216 the circumference is 432π, and 2⁄9 of 432π is exactly 96π.

Common traps

• Using the whole circumference 2πr = 96π instead of the 2⁄9 slice, which gives r = 48.
• Reaching for the area formula (θ⁄360)πr² when a length was asked, which turns the problem quadratic.
• Substituting 3.14 for π on the first line and carrying decimals through every step.

No directly related past PYQ was found.