The angle of a sector is π⁄4 radians, and the radius of the circle is 8 cm. What is the area of the sector?

Correct — D. With the angle in radians, sector area = ½ r²θ.

Square the radius: r² = 8² = 64
Multiply by θ: 64 × π⁄4 = 16π
Halve: ½ × 16π = 8π cm² → option (d)

Check in degrees: π⁄4 rad = 45°, and (45⁄360) × π × 64 = 64π ÷ 8 = 8π.

Why the others are wrong

• (a)
  16π cm² — 16π is r²θ = 64 × π⁄4 with the ½ left off. Sector area is ½ r²θ, so the true value is half of this.
• (b)
  12π cm² — 12π would need ½ × 64 × θ = 12π, so θ = 3π⁄8 radians (67.5°). The sector's angle is π⁄4, which gives 8π.
• (c)
  32π cm² — 32π is half the circle's area πr² = 64π, a semicircle. A π⁄4 sector is one-eighth of the full turn 2π, so its area is 64π ÷ 8.

A sector's area is the fraction of the circle its angle covers. In degrees the fraction is θ⁄360°, giving (θ⁄360°) × πr².

In radians the full turn is 2π, so the fraction is θ⁄2π and the area simplifies to ½ r²θ. An angle of π⁄4 is one-eighth of 2π, so the sector is one-eighth of the circle.

Key facts

• Sector area = ½ r²θ with θ in radians, or (θ⁄360°) × πr² with θ in degrees.
• Arc length = rθ with θ in radians.
• π radians = 180°, so π⁄4 = 45°.
• Sector area also equals ½ × radius × arc length.

Common traps

• Dropping the ½ in ½ r²θ, which gives 16π.
• Putting θ = π⁄4 into the degree formula (θ⁄360) × πr², which mixes units and gives about 0.44 cm².

No directly related past PYQ was found.