PreCalc 2 · Trigonometric Functions
Arc length is radius times the central angle in radians: s = rθ.
The formula
s = rθ. The angle θ must be in radians, not degrees.
If you are given degrees, convert first: multiply by π radians180°.
Worked example
r = 4 and θ = π3.
s = rθ
s = 4 · π3
s = 4π3
r = 4, θ = π3. Find s.
s = rθ
s = 4 · π3
s = 4π3
Answer: 4π3
r = 2, θ = π. Find the arc length.
s = rθ
s = 2 · π
s = 2π
Answer: 2π
r = 3, θ = π2. Find the arc length.
s = rθ
s = 3 · π2
s = 3π2
Answer: 3π2
r = 6, θ = π3. Find the arc length.
s = rθ
s = 6 · π3
s = 6π3
s = 2π
Answer: 2π
r = 10, θ = π5. Find the arc length.
s = rθ
s = 10 · π5
s = 10π5
s = 2π
Answer: 2π
r = 1, θ = 2π. Find the arc length.
s = rθ
s = 1 · 2π
s = 2π
That is the whole circumference.
Answer: 2π
r = 5 and the angle is 60°. Find the arc length.
60°1×π rad180°
θ = π3
s = rθ
s = 5 · π3
s = 5π3
Answer: 5π3
r = 8 and the angle is 90°. Find the arc length.
90°1×π rad180°
θ = π2
s = rθ
s = 8 · π2
s = 8π2
s = 4π
Answer: 4π
r = 12, θ = π6. Find the arc length.
s = rθ
s = 12 · π6
s = 12π6
s = 2π
Answer: 2π
r = 4 and the angle is 135°. Find the arc length.
135°1×π rad180°
θ = 3π4
s = rθ
s = 4 · 3π4
s = 12π4
s = 3π
Answer: 3π
r = 9 and the angle is 120°. Find the arc length.
120°1×π rad180°
θ = 2π3
s = rθ
s = 9 · 2π3
s = 18π3
s = 6π
Answer: 6π
r = 6 and the angle is 45°. Find the arc length.
45°1×π rad180°
θ = π4
s = rθ
s = 6 · π4
s = 6π4
s = 3π2
Answer: 3π2
The arc length is 4π and θ = π2. Find r.
s = rθ
r = sθ
r = 4ππ2
r = 8
Answer: 8
The arc length is 6π and r = 4. Find θ in radians.
s = rθ
θ = sr
θ = 6π4
θ = 3π2
Answer: 3π2
r = 7 and the angle is 210°. Find the arc length.
210°1×π rad180°
θ = 7π6
s = rθ
s = 7 · 7π6
s = 49π6
Answer: 49π6
The arc length is 10π3 and r = 5. Find θ.
s = rθ
θ = sr
θ = 10π35
θ = 10π15
θ = 2π3
Answer: 2π3
r = 15 and the angle is 40°. Find the arc length.
40°1×π rad180°
θ = 2π9
s = rθ
s = 15 · 2π9
s = 30π9
s = 10π3
Answer: 10π3
The arc length is 3π and r = 6. Find θ in radians.
s = rθ
θ = sr
θ = 3π6
θ = π2
Answer: π2
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