A sector of a circle having a radius of 10 cm and has a central angle of π⁄6 radians. What is the area of the sector?

Correct — A.
Formula: sector area = ½ r² θ, with θ in radians.

Square the radius: r² = 10² = 100
Halve it: ½ × 100 = 50
Multiply by θ: 50 × π⁄6 = 25π⁄3
Evaluate: 25 × 3.14159 ÷ 3 = 26.18 cm² → option (a)

Why the others are wrong

• (b)
  8.33 cm² — 8.33 = 50 × 1⁄6: the π has been dropped from π⁄6. The angle is π⁄6 ≈ 0.524 rad, not 1⁄6 rad.
• (c)
  10.47 cm² — 10.47 = 10π⁄3, two-fifths of the correct 25π⁄3. Putting r² = 100 into ½r²θ gives 26.18, not 10.47.
• (d)
  12.5 cm² — 12.5 = ½ × 100 × 0.25, which would need θ = 0.25 rad. But π⁄6 ≈ 0.524 rad, so the true area is a little over twice 12.5.

A sector is the slice of a circle between two radii. Its area is the circle's area scaled by the angle's share of a full turn.

In radians a full turn is 2π, so area = (θ ⁄ 2π) × πr² = ½ r² θ.

Check in degrees: π⁄6 = 30°, and 30⁄360 × π × 10² = 100π⁄12 = 25π⁄3 ≈ 26.18 cm², the same value.

Key facts

• Sector area = ½ r² θ with θ in radians, or (θ ⁄ 360°) × πr² with θ in degrees.
• Arc length = r θ with θ in radians.
• π⁄6 radians = 30°, one-twelfth of a full turn.
• 25π⁄3 ≈ 26.18.

Common traps

• Dropping π from π⁄6 and using 1⁄6 as the angle, which gives 8.33.
• Using the arc-length formula rθ (≈ 5.24 cm) for an area question.
• Forgetting the ½ in ½r²θ, which gives 52.36 cm².

No directly related past PYQ was found.