Arc Length Formula

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

The Formula

s=rθ (radians)ors=θ360°⋅2πr (degrees)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°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 1010 and central angle π3\frac{\pi}{3} radians: s=10⋅π3=10π3≈10.47 unitss = 10 \cdot \frac{\pi}{3} = \frac{10\pi}{3} \approx 10.47 \text{ units}

Notation

ss for arc length, rr for radius, θ\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°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=rθs = r\theta for θ\theta in radians; general arc length for parametric curve γ(t)=(x(t),y(t))\gamma(t) = (x(t), y(t)), t∈[a,b]t \in [a,b]: s=∫abx′(t)2+y′(t)2 dts = \int_a^b \sqrt{x'(t)^2 + y'(t)^2}\,dt

Worked Examples

Example 1

easy
A circle has radius 66 cm. Find the arc length intercepted by a central angle of 60°60°.

Answer

s=2π≈6.28s = 2\pi \approx 6.28 cm

First step

1
Step 1: Write the arc length formula using degrees: s=θ360°×2πrs = \frac{\theta}{360°} \times 2\pi r.

Full solution

  1. 2
    Step 2: Substitute θ=60°\theta = 60° and r=6r = 6 cm: s=60360×2π(6)s = \frac{60}{360} \times 2\pi(6).
  2. 3
    Step 3: Simplify the fraction: 60360=16\frac{60}{360} = \frac{1}{6}, so s=16×12πs = \frac{1}{6} \times 12\pi.
  3. 4
    Step 4: Compute: s=2π≈6.28s = 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 2π(6)=12π2\pi(6) = 12\pi, giving 2π2\pi cm.

Example 2

medium
A circle has radius 55 m. Find the arc length subtended by a central angle of 3π4\frac{3\pi}{4} radians.

Example 3

medium
A circle has radius 1010 cm. Find the arc length for a central angle of 135°135° in terms of π\pi.

Common Mistakes

  • Using πr2\pi r^2 (area) instead of 2πr2\pi r (circumference) for the whole — arc length scales the circumference, not the area.
  • Forgetting the θ360°\frac{\theta}{360°} fraction and giving the full circumference — only the angle's share counts.
  • Leaving the answer in degrees — arc length is a distance in length units, not degrees.

Why This Formula Matters

It is the first place students scale a whole quantity by an angle fraction, the same move that defines sector area and the radian; mixing up distance (arc length) with degrees (the angle itself) is a persistent error this concept must fix. Recognizing it by "Am I asked for a length along the circle's edge (not an angle and not an enclosed area)?" — rather than by familiar numbers — is what lets a student tell it apart from circumference and sector area and central angle in a mixed problem set.

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°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?

ss for arc length, rr for radius, θ\theta for central angle

Why is the Arc Length formula important in Math?

It is the first place students scale a whole quantity by an angle fraction, the same move that defines sector area and the radian; mixing up distance (arc length) with degrees (the angle itself) is a persistent error this concept must fix. Recognizing it by "Am I asked for a length along the circle's edge (not an angle and not an enclosed area)?" — rather than by familiar numbers — is what lets a student tell it apart from circumference and sector area and central angle in a mixed problem set.

What do students get wrong about Arc Length?

The procedure for arc length is the easy part; the trap is using πr2\pi r^2 (area) instead of 2πr2\pi r (circumference) for the whole. Asking "Am I asked for a length along the circle's edge (not an angle and not an enclosed area)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Arc Length formula?

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

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