Radian measure; arc length and sector area practice questions.

A sector of a circle has radius 6 cm and central angle π/4 radians. Calculate the arc length of the sector.

• 3π cm
• 3π/2 cm
• 9π/2 cm
• 3π/4 cm

Answer: 3π/2 cm

Arc length s = rθ = 6 × π/4 = 3π/2 cm; using ½r²θ instead gives the sector area 9π/2, a common mix-up.

A sector of a circle of radius 4 cm has area 12 cm². Determine the central angle in radians.

• 1.5
• 3
• 0.75
• 6

Answer: 1.5

A = ½r²θ gives 12 = ½ × 16 × θ, so θ = 12 ÷ 8 = 1.5 radians.

Convert 150° to radians.

• 3π/4
• 2π/3
• 5π/12
• 5π/6

Answer: 5π/6

Multiply by π/180: 150 × π/180 = 5π/6 radians.

A sector has radius 4 cm and central angle 1.5 radians. Calculate the perimeter of the sector.

• 6 cm
• 12 cm
• 14 cm
• 10 cm

Answer: 14 cm

Perimeter = two radii + arc = 2(4) + 4 × 1.5 = 8 + 6 = 14 cm; forgetting the two radii gives only 6 cm.

Which statement gives the definition of an angle of one radian?

• The angle at the centre of a circle subtended by an arc equal in length to the radius
• The angle at the centre of a circle subtended by an arc equal in length to the diameter
• The angle equal to 1/360 of a full turn
• The angle at the centre of a circle subtended by a chord equal in length to the radius

Answer: The angle at the centre of a circle subtended by an arc equal in length to the radius

One radian is defined by arc length = radius, which is why s = rθ holds only in radians.

A chord subtends an angle of π/3 radians at the centre of a circle of radius 6 cm. Determine the exact area of the minor segment cut off by the chord.

• 6π cm²
• (6π − 9√3) cm²
• (6π + 9√3) cm²
• 9√3 cm²

Answer: (6π − 9√3) cm²

Segment = sector − triangle = ½ × 6² × π/3 − ½ × 6² × sin(π/3) = 6π − 9√3 cm².