The chord ST of a circle is equal to the radius r of the circle. Find the length of arc ST

Analyze the question: Chord ST has the same length as the radius r. We need the length of the (minor) arc ST.

Concept: If O is the centre, then OS = OT = r (radii). Since ST = r as well, triangle OST has all three sides equal, so it is equilateral.

Solution:

1. Triangle OST is equilateral, so each angle is 60°.
2. The angle subtended at the centre by the chord is θ = 60°.
3. Arc length = (θ/360°) × 2πr

Arc ST = (60/360) × 2πr = (1/6) × 2πr = πr/3

Check using radians: 60° = π/3 rad, and arc length = rθ = r × π/3 = πr/3. ✔

Answer: The length of arc ST is πr/3, which is option A.

Tip: When a chord equals the radius, think "equilateral triangle, 60° at the centre." Then the arc is simply 1/6 of the circumference (2πr ÷ 6 = πr/3). Option D (πr/6) is a trap that comes from forgetting the 2 in 2πr.