Arc Length Calculator ($L$)

Calculate the length of the arc ($L$) along the circumference of a circle, determined by its radius ($r$) and the central angle ($\theta$). Enter your values below and select whether the angle is in degrees or radians.

Formulas for Arc Length

The arc length is defined as a fraction of the circle's total circumference. The formula used depends on whether the central angle ($\theta$) is measured in radians or degrees.

1. Arc Length using Radians ($\theta$)

The simplest and most fundamental relationship: the arc length ($L$) is the product of the radius and the angle (in radians).

This is the preferred formula for scientific and engineering calculations because of its simplicity.

2. Arc Length using Degrees ($\theta^\circ$)

The formula for degrees explicitly shows the ratio of the partial angle to the full circle:

Radians Conversion

If you start with degrees, you must first convert the angle to radians to use the simpler formula:

Arc Length vs. Chord Length

It is important to distinguish the arc length from the chord length (which is the straight-line distance between the two endpoints of the arc). The arc length is always greater than the chord length, except when the angle is $0^\circ$ or $360^\circ$, where they are equal.

Frequently Asked Questions (FAQ)

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L = r \theta

L = 2\pi r \left( \frac{\theta}{360^\circ} \right)

\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180^\circ}

L = C \frac{\theta}{360^\circ}

\mathcal{A} = \frac{1}{2} r L

$L = 2\pi r \left( \frac{\theta}{360^\circ} \right)$

$\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180^\circ}$