At what angle is the arc length half the perimeter of the circle?

Correct — D. An arc takes the same fraction of the circumference that its central angle takes of 360°.

Arc = (θ ⁄ 360°) × 2πr
Circumference = 2πr, so half of it = πr

Set the two equal: (θ ⁄ 360°) × 2πr = πr
Cancel πr from both sides: 2θ ⁄ 360° = 1
θ ⁄ 360° = 1⁄2
θ = 180° → option (d)

The radius cancels, so no value for r was ever needed.

Why the others are wrong

• (a)
  120° — 120° cuts 120 ⁄ 360 = one third of the circumference, an arc of 2πr⁄3. That falls short of the πr the question demands.
• (b)
  270° — 270° cuts three quarters of the circumference, 3πr⁄2. That is half as much again as πr, so the arc overshoots rather than halves.
• (c)
  90° — 90° gives a quarter of the circumference, πr⁄2 — half the arc wanted. A right angle halves the semicircle, not the circle.

Everything about a sector runs off one fraction, θ ⁄ 360°.

Arc length is that fraction of 2πr, and sector area is the same fraction of πr². So a 180° angle gives half the circumference and half the area, and a 90° angle gives a quarter of each.

The chord is the exception: it is not proportional to θ, which is why a 180° chord is the diameter 2r while a 90° chord is r√2, not half of it.

Key facts

• Arc length = (θ ⁄ 360°) × 2πr, so the arc is the same fraction of the circumference as θ is of 360°.
• A 180° arc is a semicircular arc measuring πr, exactly half of the circumference 2πr.
• Sector area = (θ ⁄ 360°) × πr², so a 180° sector is also half the circle's area.
• The radius cancels in this equation, so the answer is 180° for a circle of any size.

Common traps

• Reading 'perimeter of the circle' as a semicircle's boundary, which is πr + 2r.
• Answering 90° because the word 'half' suggests a right angle.
• Hunting for the radius, which cancels out and never had to be known.

No directly related past PYQ was found.