Arc length is the distance along a curved part of a circle, measured in units of length. An arc with central angle θ in a circle of radius r is θ360 of the circumference: s = θ360 × 2π r when θ is in degrees, or simply s = rθ when θ is in radians.
In Spanish
longitud de arco
Also called
length of an arc, arc measure in length, curved distance
60° is 16 of a full turn, so the arc is 16 of the circumference 12π, which is 2π.
Formal definition and notation
For a circle of radiusr and a central angleθ, the intercepted arc has lengths = rθ when θ is measured in radians, which is equivalent to s = θ360° · 2π r when θ is measured in degrees. Arc length is proportional to the radius for a fixed angle, and the radian measure of the angle is the constant of proportionalitysr. In calculus, the arc length of a curve y = f(x) is extended to any smooth curve with an integral.
Symbol
Read it as
Meaning
s
s
the arc length
θ
theta
the central angle that cuts off the arc
Key facts
Arc length is a length (cm, ft); arc measure is an angle (degrees). A 60° arc is 60° in every circle, but its length depends on the radius.
A full circle is 360° and its arc length is the circumference 2π r; a semicircle has arc length π r.
In radians the formula is just s = rθ, which is why radians are defined as arc length divided by radius.
Doubling the radius doubles the arc length for the same angle.
To find the angle from an arc, solve for θ: θ = s2π r × 360.
Arc length formulas
Arc length formulas
s = θ360 × 2π r (degrees), s = rθ (radians)
s
arc length, in the same unit as the radius
r
radius of the circle
θ
central angle
Both formulas say the same thing: the arc is the fraction of the circle that the angle covers. 180 degrees equals π radians.
Arc Length examples, step by step
Example 1: A quarter circle
Warm-up
Find the length of a 90° arc in a circle with radius 10 cm. Give the exact answer and a decimal.
90° is 90360 = 14 of the circle.
Circumference: 2π × 10 = 20π.
s = 14 × 20π = 5π ≈ 15.71.
Answer: The arc is 5π cm, about 15.71 cm.
Example 2: An angle in radians
Core
A circle has radius 8 m. Find the arc length cut off by a central angle of 3π4 radians.
In radians, use s = rθ.
s = 8 × 3π4.
s = 6π ≈ 18.85.
Answer: The arc is 6π m, about 18.85 m.
Example 3: The tip of a minute hand
Test-style
The minute hand of a clock is 15 cm long. How far does its tip travel in 20 minutes? Round to the nearest tenth.
In 60 minutes the hand turns 360°, so in 20 minutes it turns 2060 × 360 = 120°.
The tip moves along a circle with radius 15: circumference 30π.
s = 120360 × 30π = 10π.
10π ≈ 31.4.
Answer: The tip travels 10π cm, about 31.4 cm.
Common mistakes
✗ MistakeUsing the diameter in 2π r: for r = 9 and 40°, s = 40360 × 2π × 18 = 4π.
✓ InsteadUse the radius: s = 40360 × 2π × 9 = 2π.
2π r already doubles the radius. Putting in the diameter doubles it twice.
✗ MistakeUsing s = rθ with degrees: r = 6, 60°, so s = 360.
✓ InsteadConvert to radians first: 60° = π3, so s = 6 × π3 = 2π.
s = rθ is true only when θ is in radians.
✗ MistakeGiving the arc length of a 60 degree arc as 60.
✓ Instead60° is the arc measure. In a circle of radius 6 the arc length is 2π ≈ 6.28.
Arc measure is an angle in degrees; arc length is a distance in units.
Not to be confused with
Arc Lengthvs.Arc measure
Arc measure is the central angle in degrees (a 60° arc); arc length is the distance along the arc (2π cm when r = 6).
A chord is the straight segment between the arc’s endpoints. It is always shorter than the arc.
Arc Lengthvs.Sector area
Sector area is the space inside the slice: θ360 × π r2. Arc length is only the curved edge: θ360 × 2π r.
Quick check: 3 questions
Question 1 of 3
What is the length of a 40° arc in a circle with radius 9?
Show answer and explanation
Answer: C.40360 × 2π × 9 = 19 × 18π = 2π. 4π uses the diameter as r, π forgets the 2 in 2π r, and 9π is half the circumference.
Question 2 of 3
A circle has radius 5. What arc length does a central angle of 2 radians cut off?
Show answer and explanation
Answer: A. In radians s = rθ = 5 × 2 = 10. 2.5 divides, 7 adds, and 10π wrongly mixes in π, which is only needed with degrees.
Question 3 of 3
An arc of length 4π is in a circle of radius 12. What is its central angle?
Show answer and explanation
Answer: B. Fraction of the circle: 4π24π = 16, and 16 × 360 = 60°. 30° uses the diameter, 120° uses the radius instead of the circumference, and 4° confuses length with angle.
Arc Length: frequently asked questions
What is the formula for arc length?
With the central angle in degrees, s = θ360 × 2π r: the fraction of the circle times the circumference. With the angle in radians, s = rθ. For a radius of 4 and an angle of 45° (π4 radians), both give s = π ≈ 3.14.
What is the difference between arc length and arc measure?
Arc measure is an angle: the number of degrees in the central angle, like 90°. Arc length is a distance: how far you would walk along the curve, like 15.7 cm. Two circles can both have a 90° arc, but the bigger circle’s arc is longer.
Why is the arc length formula simpler in radians?
A radian is defined so that an angle of 1 radian cuts off an arc exactly one radius long. So an angle of θ radians cuts off θ radii of arc, which is s = rθ. In degrees you need the extra factor 2π360 to convert.
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