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Math

Arc Length

4 min readMedium10-question checkpoint

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.

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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:

\text{Arc length} = \frac{\theta}{360} \cdot 2\pi r

The fraction $\frac{\theta}{360}$ is the angle's share of the full circle. That fraction × the full circumference = the arc length.

The radians formula:

\text{Arc length} = r \theta

Where $\theta$ is in radians. (Radians are the more 'natural' unit for these formulas — no $360$ to divide by.)

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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 $\theta = 2\pi$)

Converting between degrees and radians:

$360° = 2\pi$ radians (full circle).
$180° = \pi$ radians.
$90° = \pi/2$ radians.
$60° = \pi/3$ radians.
$45° = \pi/4$ radians.

General conversion: degrees → radians, multiply by $\pi/180$ . Radians → degrees, multiply by $180/\pi$ .

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 $2\pi r$ to get the arc length.

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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°	$\pi$	$1/2$	$12\pi$

Common SAT setups:

"A circle of radius 10 has a central angle of 60°. Find the arc length." → $\frac{60}{360} \cdot 2\pi \cdot 10 = \frac{1}{6} \cdot 20\pi = \frac{10\pi}{3}$ .
"A wheel of radius 20 cm rolls one full revolution. How far does it travel?" → one revolution = full circumference = $2\pi \cdot 20 = 40\pi$ cm.
"An arc of length $\frac{5\pi}{2}$ subtends a central angle of 90° in a circle. Find the radius." → $\frac{5\pi}{2} = \frac{1}{4} \cdot 2\pi r \Rightarrow r = 5$ .

Sector area is the 2D analog. Same fraction of the full circle's area:

\text{Sector area} = \frac{\theta}{360} \cdot \pi r^2

In radians: $\frac{1}{2} r^2 \theta$ .

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).

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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.

A circle has center (3, -2) and radius 5. Which of the following is the equation of the circle?

(x + 3)² + (y - 2)² = 25

(x - 3)² + (y - 2)² = 25

(x + 3)² + (y + 2)² = 5

(x - 3)² + (y + 2)² = 25

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

An arc of length $5\pi$ subtends a central angle of $\frac{\pi}{4}$ radians in a circle. What is the radius?

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Common pitfalls

Mixing radians and degrees in one formula

$\frac{\theta}{360} \cdot 2\pi r$ is for $\theta$ in DEGREES. $r\theta$ is for $\theta$ in RADIANS. If you mix the two (e.g., $\theta$ in radians divided by 360), you'll be off by $\pi/180$ .

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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

Arc length is one-dimensional ( $\frac{\theta}{360} \cdot 2\pi r$ ). Sector area is two-dimensional ( $\frac{\theta}{360} \cdot \pi r^2$ ). Don't substitute one for the other.

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Computing fraction of 100 instead of 360

The full circle is 360°, not 100°. Even if a question feels like a percent, the angle fraction uses 360 in the denominator: $\frac{\theta}{360}$ , not $\frac{\theta}{100}$ .

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Key takeaways

Arc length (degrees) = $\frac{\theta}{360} \cdot 2\pi r$ . Arc length (radians) = $r\theta$ .
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The arc is a fraction of the circumference — that fraction is the angle's share of $360°$ (or $2\pi$ radians).
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Convert: degrees × $\pi/180$ = radians. Radians × $180/\pi$ = degrees.
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Sector area uses the same fraction: $\frac{\theta}{360} \cdot \pi r^2$ (or $\frac{1}{2}r^2\theta$ in radians).
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Arc length is along the curve; chord is the straight line between endpoints. They're not the same.

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Where this lesson fits

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Arc Length

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