A circular jogging track has a radius of 50 m. A runner covers a curved path that subtends an angle of 120° at the center. What is the length of the arc she runs?

Correct — A.

Arc length = (θ⁄360°) × 2πr
= (120⁄360) × 2π × 50
= (1⁄3) × 100π
= 100π⁄3 m → option (a).

Check: 120° is one third of 360°, so the arc is a third of the 100π m track.

Why the others are wrong

• (b)
  150π⁄3 m — 150π⁄3 m is 50π m, the arc for 180°, half the 100π m track. 120° is a third of the circle, not half.
• (c)
  200π⁄3 m — 200π⁄3 m is the major arc, (240⁄360) × 100π, the part of the track she does not run. Putting the diameter 100 m in place of the radius gives the same wrong value.
• (d)
  200π⁄3 m — Printed identically to (c). 200π⁄3 m is double the right arc: the 240° major arc, or the formula run with the diameter instead of the radius.

An arc takes the same fraction of the circumference as its angle takes of 360°. The track's circumference is 2π × 50 = 100π m, and 120° is a third of 360°, so the arc is a third of 100π.

In radians the same rule reads ℓ = rθ: 120° = 2π⁄3, and 50 × 2π⁄3 = 100π⁄3.

Key facts

• Arc length = (θ⁄360°) × 2πr.
• In radians, arc length = rθ, and 120° = 2π⁄3 rad.
• Sector area = (θ⁄360°) × πr², which also equals ½ × r × arc length.
• 100π⁄3 m ≈ 104.7 m.

Common traps

• Putting the diameter, 100 m, into 2πr. That doubles the arc to 200π⁄3 m.
• Reaching for the sector-area formula (θ⁄360°) × πr², which gives 2500π⁄3 m², an area rather than a length.

No directly related past PYQ was found.