IB AA · Unit 2 · 18A Radian Measure
Arc length is radius times the angle in radians. Sector area is half the radius squared times that angle.
Arc length and sector area
l = rθ
A = 12 r² θ
θ is in radians. A full turn is 2π radians.
Degrees first
If the angle is in degrees, convert before you substitute.
Multiply by π radians180°.
The marked angle is the shaded angle. θ is already in radians.
l = rθ
l = 9 × 1.2
l = 10.8 cm
A = 12 r² θ
A = 12 × 9² × 1.2
A = 48.6 cm²
Answer: arc 10.8 cm, area 48.6 cm²
l = rθ
l = 7 × 0.5
l = 3.5 cm
A = 12 r² θ
A = 12 × 7² × 0.5
A = 12.25 cm²
Answer: arc 3.5 cm, area 12.25 cm²
l = rθ
l = 11 × 1.4
l = 15.4 cm
A = 12 r² θ
A = 12 × 11² × 1.4
A = 84.7 cm²
Answer: arc 15.4 cm, area 84.7 cm²
l = rθ
l = 4 × 2.5
l = 10 cm
A = 12 r² θ
A = 12 × 4² × 2.5
A = 20 cm²
Answer: arc 10 cm, area 20 cm²
l = rθ
l = 15 × 0.4
l = 6 cm
A = 12 r² θ
A = 12 × 15² × 0.4
A = 45 cm²
Answer: arc 6 cm, area 45 cm²
l = rθ
l = 8 × 1.7
l = 13.6 cm
A = 12 r² θ
A = 12 × 8² × 1.7
A = 54.4 cm²
Answer: arc 13.6 cm, area 54.4 cm²
The shaded angle is 2π minus the marked angle.
The shaded angle is one full turn minus the marked angle.
θ = 2π − 1.2
l = rθ
l = 5(2π − 1.2)
l = 10π − 6
l ≈ 25.4 cm
A = 12 r² θ
A = 12 × 5² × (2π − 1.2)
A = 25π − 15
A ≈ 63.5 cm²
Answer: arc 25.4 cm, area 63.5 cm²
The shaded angle is one full turn minus the marked angle.
θ = 2π − 0.9
l = rθ
l = 8(2π − 0.9)
l = 16π − 7.2
l ≈ 43.1 cm
A = 12 r² θ
A = 12 × 8² × (2π − 0.9)
A = 64π − 28.8
A ≈ 172 cm²
Answer: arc 43.1 cm, area 172 cm²
The shaded angle is one full turn minus the marked angle.
θ = 2π − 1.5
l = rθ
l = 6(2π − 1.5)
l = 12π − 9
l ≈ 28.7 cm
A = 12 r² θ
A = 12 × 6² × (2π − 1.5)
A = 36π − 27
A ≈ 86.1 cm²
Answer: arc 28.7 cm, area 86.1 cm²
The shaded angle is one full turn minus the marked angle.
θ = 2π − 0.7
l = rθ
l = 10(2π − 0.7)
l = 20π − 7
l ≈ 55.8 cm
A = 12 r² θ
A = 12 × 10² × (2π − 0.7)
A = 100π − 35
A ≈ 279 cm²
Answer: arc 55.8 cm, area 279 cm²
The shaded angle is one full turn minus the marked angle.
θ = 2π − 1.1
l = rθ
l = 7(2π − 1.1)
l = 14π − 7.7
l ≈ 36.3 cm
A = 12 r² θ
A = 12 × 7² × (2π − 1.1)
A = 49π − 26.95
A ≈ 127 cm²
Answer: arc 36.3 cm, area 127 cm²
The shaded angle is one full turn minus the marked angle.
θ = 2π − 2
l = rθ
l = 4(2π − 2)
l = 8π − 8
l ≈ 17.1 cm
A = 12 r² θ
A = 12 × 4² × (2π − 2)
A = 16π − 16
A ≈ 34.3 cm²
Answer: arc 17.1 cm, area 34.3 cm²
Convert to radians first, then use the formulas.
72°1×π rad180°
Degrees go on the bottom of the conversion so the degrees cancel.
θ = 2π5
l = rθ
l = 10 × 2π5
l = 4π
l ≈ 12.6 cm
A = 12 r² θ
A = 12 × 10² × 2π5
A = 20π
A ≈ 62.8 cm²
Answer: arc 4π cm, about 12.6 cm; area 20π cm², about 62.8 cm²
150°1×π rad180°
Degrees go on the bottom of the conversion so the degrees cancel.
θ = 5π6
l = rθ
l = 6 × 5π6
l = 5π
l ≈ 15.7 cm
A = 12 r² θ
A = 12 × 6² × 5π6
A = 15π
A ≈ 47.1 cm²
Answer: arc 5π cm, about 15.7 cm; area 15π cm², about 47.1 cm²
120°1×π rad180°
Degrees go on the bottom of the conversion so the degrees cancel.
θ = 2π3
l = rθ
l = 8 × 2π3
l = 16π3
l ≈ 16.8 cm
A = 12 r² θ
A = 12 × 8² × 2π3
A = 64π3
A ≈ 67.0 cm²
Answer: arc 16π3 cm, about 16.8 cm; area 64π3 cm², about 67.0 cm²
80°1×π rad180°
Degrees go on the bottom of the conversion so the degrees cancel.
θ = 4π9
l = rθ
l = 9 × 4π9
l = 4π
l ≈ 12.6 cm
A = 12 r² θ
A = 12 × 9² × 4π9
A = 18π
A ≈ 56.5 cm²
Answer: arc 4π cm, about 12.6 cm; area 18π cm², about 56.5 cm²
210°1×π rad180°
Degrees go on the bottom of the conversion so the degrees cancel.
θ = 7π6
l = rθ
l = 5 × 7π6
l = 35π6
l ≈ 18.3 cm
A = 12 r² θ
A = 12 × 5² × 7π6
A = 175π12
A ≈ 45.8 cm²
Answer: arc 35π6 cm, about 18.3 cm; area 175π12 cm², about 45.8 cm²
30°1×π rad180°
Degrees go on the bottom of the conversion so the degrees cancel.
θ = π6
l = rθ
l = 14 × π6
l = 7π3
l ≈ 7.33 cm
A = 12 r² θ
A = 12 × 14² × π6
A = 49π3
A ≈ 51.3 cm²
Answer: arc 7π3 cm, about 7.33 cm; area 49π3 cm², about 51.3 cm²
Want to go over a radian or a trig value together?
Book a Session with Amy