Arc Length

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.)

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.

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).