Arc Length Formula

An arc is a portion of a circle’s circumference. Its length is L = (θ/360)·2πr for a central angle θ in degrees, or simply L = rθ when θ is in radians — the fraction of the full circle times the circumference.

Arc Length formulas

Arc length (degrees): $L = \dfrac{\theta}{360}\cdot 2\pi r$
Arc length (radians): $L = r\theta$

where r = radius, θ = central angle (degrees), L = arc length

Arc length (degrees)

L = \dfrac{\theta}{360}\cdot 2\pi r

Worked example

A 90° arc on a circle of radius 6 is one quarter of the circumference: 3π ≈ 9.42.

L = \dfrac{90}{360}\cdot 2\pi\cdot 6 = \tfrac{1}{4}\cdot 12\pi = 3\pi \approx 9.42

Arc length (radians)

L = r\theta

Interactive arc length explorer

Drag a slider (or type a value, or use the arrow keys) and watch the diagram, the results, and the full step-by-step substitution update live.

Radius r

Central angle θ θ

Arc length L 9.42units

Step-by-step substitution

L = (θ/360) · 2πr = (90/360) · 2π · 6 = 9.42

Classroom use: arc length formulas

Who it's for: Grades 9–12 · High-school geometry students working on circle measurement, and precalculus students about to meet radian measure.

Learning objectives

After using this page, students will be able to:

Compute arc length from a radius and a central angle in degrees.
Express arc length as a fraction θ/360 of the full circumference.
Explain why arc length scales proportionally with the radius.

Suggested classroom activity

Set the angle to 90° and ask what fraction of the circumference the arc must be before revealing the number. Then vary the radius at fixed angle: the proportionality that motivates radian measure is exactly what students see, which makes the later definition feel inevitable.

Standards alignment

Aligned to the Common Core State Standards for Mathematics. Codes are given exactly as published so they can be matched against state crosswalks.
Code	Standard
`HSG.C.B.5`	Derive the arc-length and sector-area relationships using similarity.
`7.G.B.4`	Know and use the formulas for the area and circumference of a circle.

Free to use and adapt in class under the terms on our license page — no account, no sign-up, no ads. Content last reviewed August 2026.

Related & next steps

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Frequently asked questions

What is the arc length formula?

Arc length is L = (θ/360)·2πr with θ in degrees, or L = rθ with θ in radians. It is the fraction θ/360 of the full circumference 2πr.

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