Arc length is just a fraction of the circle's circumference — the fraction equal to the central angle's share of 360°. Memorize the two formulas (degrees vs radians) and these problems become arithmetic.
Arc length & sector area — fraction-of-circle pattern
Measure
Degrees formula
Radians formula
Arc length
$\frac{\theta}{360} \cdot 2\pi r$
$r\theta$
Sector area
$\frac{\theta}{360} \cdot \pi r^2$
$\frac{1}{2}r^2\theta$
Full circumference
$2\pi r$ (when $\theta = 360°$)
$2\pi r$ (when $\theta = 2\pi$)
Full area
$\pi r^2$ (when $\theta = 360°$)
$\pi r^2$ (when $\theta = 2\pi$)
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Common central angles — fraction of circle
Angle (degrees)
Angle (radians)
Fraction of circle
Arc with $r=12$
30°
$\pi/6$
$1/12$
$2\pi$
45°
$\pi/4$
$1/8$
$3\pi$
60°
$\pi/3$
$1/6$
$4\pi$
90°
$\pi/2$
$1/4$
$6\pi$
120°
$2\pi/3$
$1/3$
$8\pi$
180°
$\pi$
$1/2$
$12\pi$
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Quick check
Pick the right formula based on whether the angle is in degrees or radians. Or compute the angle's fraction of the full circle and apply it to the circumference.
An arc of a circle with radius 10 subtends a central angle of 72°. What is the length of the arc? (Use π ≈ 3.14)
Worked examples
Example 1
A circle has radius 12. What is the length of the arc subtended by a central angle of 30°?
Example 2
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Common pitfalls
Mixing radians and degrees in one formula
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Forgetting that arc length is curved, not straight
The chord is the straight line between the arc's endpoints — DIFFERENT from arc length. Arc is always longer than its chord. SAT trap answers sometimes give the chord.
Confusing arc length with sector area
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Computing fraction of 100 instead of 360
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Key takeaways
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Arc length is along the curve; chord is the straight line between endpoints. They're not the same.
Tracks your progress across lessons.
Try it yourself
5 practice questions on Arc Length, drawn from the question bank. The tutor is one click away if you get stuck.
An arc is a piece of a circle's circumference. The arc length depends on (a) the radius and (b) the central angle that subtends it.
The degrees formula:
Arc length=360θ⋅2πr
The fraction 360θ is the angle's share of the full circle. That fraction × the full circumference = the arc length.
The radians formula:
Arc length=rθ
Where θ is in radians. (Radians are the more 'natural' unit for these formulas — no 360 to divide by.)
Converting between degrees and radians:
360°=2π radians (full circle).
180°=π radians.
90°=π/2 radians.
60°=π/3 radians.
45°=π/4 radians.
General conversion: degrees → radians, multiply by π/180. Radians → degrees, multiply by 180/π.
The 'fraction of circle' shortcut. For common angles, the fraction is:
30° → 1/12
45° → 1/8
60° → 1/6
72° → 1/5
90° → 1/4
120° → 1/3
180° → 1/2
Multiply that fraction by 2πr to get the arc length.
Common SAT setups:
"A circle of radius 10 has a central angle of 60°. Find the arc length." → 36060⋅2π⋅10=61⋅20π=310π.
"A wheel of radius 20 cm rolls one full revolution. How far does it travel?" → one revolution = full circumference = 2π⋅20=40π cm.
"An arc of length 25π subtends a central angle of 90° in a circle. Find the radius." → 25π=41⋅2πr⇒r=5.
Sector area is the 2D analog. Same fraction of the full circle's area:
Sector area=360θ⋅πr2
In radians: 21r2θ.
Don't confuse arc length with chord length. Arc length is along the curve. Chord length is the straight-line distance between the arc's endpoints. They're different — and an arc is always longer than its chord (for non-zero angle).
An arc of length 5π subtends a central angle of 4π radians in a circle. What is the radius?
360θ⋅2πr is for θ in DEGREES. rθ is for θ in RADIANS. If you mix the two (e.g., θ in radians divided by 360), you'll be off by π/180.
Arc length is one-dimensional (360θ⋅2πr). Sector area is two-dimensional (360θ⋅πr2). Don't substitute one for the other.
The full circle is 360°, not 100°. Even if a question feels like a percent, the angle fraction uses 360 in the denominator: 360θ, not 100θ.