Arc Length Formula

The Formula

s = r\theta \text{ (radians)} \quad \text{or} \quad s = \frac{\theta}{360°} \cdot 2\pi r \text{ (degrees)}

When to use: Imagine walking along a circular track but only covering a portion of the full loop. The arc length is how far you actually walked. If you walk a quarter of the circle (90°), you cover a quarter of the circumference. The fraction of the full circle you cover determines the fraction of the circumference you walk.

Quick Example

A circle with radius 10 and central angle \frac{\pi}{3} radians: s = 10 \cdot \frac{\pi}{3} = \frac{10\pi}{3} \approx 10.47 \text{ units}

Notation

s for arc length, r for radius, \theta for central angle

What This Formula Means

The distance along a portion of a circle's circumference, determined by the central angle and the radius.

Imagine walking along a circular track but only covering a portion of the full loop. The arc length is how far you actually walked. If you walk a quarter of the circle (90°), you cover a quarter of the circumference. The fraction of the full circle you cover determines the fraction of the circumference you walk.

Formal View

s = r\theta for \theta in radians; general arc length for parametric curve \gamma(t) = (x(t), y(t)), t \in [a,b]: s = \int_a^b \sqrt{x'(t)^2 + y'(t)^2}\,dt

Worked Examples

Example 1

easy
A circle has radius 6 cm. Find the arc length intercepted by a central angle of 60°.

Solution

  1. 1
    Step 1: Write the arc length formula using degrees: s = \frac{\theta}{360°} \times 2\pi r.
  2. 2
    Step 2: Substitute \theta = 60° and r = 6 cm: s = \frac{60}{360} \times 2\pi(6).
  3. 3
    Step 3: Simplify the fraction: \frac{60}{360} = \frac{1}{6}, so s = \frac{1}{6} \times 12\pi.
  4. 4
    Step 4: Compute: s = 2\pi \approx 6.28 cm.

Answer

s = 2\pi \approx 6.28 cm
Arc length is the fraction of the full circumference determined by the central angle. A 60° angle is one-sixth of 360°, so the arc is one-sixth of the full circumference 2\pi(6) = 12\pi, giving 2\pi cm.

Example 2

medium
A circle has radius 5 m. Find the arc length subtended by a central angle of \frac{3\pi}{4} radians.

Common Mistakes

  • Using degrees in the radian formula (s = r\theta) without converting
  • Confusing arc length (a distance) with arc measure (an angle in degrees)
  • Forgetting to use the correct angle unit for the chosen formula

Why This Formula Matters

Used in engineering (gear teeth, pulley belts), navigation (latitude/longitude), and physics (angular displacement).

Frequently Asked Questions

What is the Arc Length formula?

The distance along a portion of a circle's circumference, determined by the central angle and the radius.

How do you use the Arc Length formula?

Imagine walking along a circular track but only covering a portion of the full loop. The arc length is how far you actually walked. If you walk a quarter of the circle (90°), you cover a quarter of the circumference. The fraction of the full circle you cover determines the fraction of the circumference you walk.

What do the symbols mean in the Arc Length formula?

s for arc length, r for radius, \theta for central angle

Why is the Arc Length formula important in Math?

Used in engineering (gear teeth, pulley belts), navigation (latitude/longitude), and physics (angular displacement).

What do students get wrong about Arc Length?

Make sure the angle and formula match—use radians with s = r\theta and degrees with s = \frac{\theta}{360} \cdot 2\pi r.

What should I learn before the Arc Length formula?

Before studying the Arc Length formula, you should understand: circumference, central angle.

← Back to Arc Length See All Examples Practice Problems