Arc Length Calculator – Circle Arc with Radius & Angle
A circular arc is a portion of a circle’s circumference.
If a circle has radius \(r\) and an arc subtends a central angle \(\theta\),
then the arc length depends only on \(r\) and \(\theta\).
1) Arc length in radians
When \(\theta\) is measured in radians, the arc length formula is:
\[
s = r\theta.
\]
This is the cleanest form because the radian measure is defined so that
“angle = arc length / radius”.
2) Arc length in degrees
If \(\theta\) is measured in degrees, use the proportion of the full circle:
\[
s = \frac{\theta}{360}\cdot 2\pi r.
\]
This works because \(2\pi r\) is the full circumference and \(\theta/360\) is the fraction of the circle.
3) Converting degrees ↔ radians
\[
\theta_{\text{rad}}=\theta_{\deg}\cdot\frac{\pi}{180},
\qquad
\theta_{\deg}=\theta_{\text{rad}}\cdot\frac{180}{\pi}.
\]
4) Solving for the angle from arc length
If you know \(s\) and \(r\), solve the radian formula for \(\theta\):
\[
\theta=\frac{s}{r}\quad(\text{radians}).
\]
You can then convert \(\theta\) to degrees using \(\theta_{\deg}=\theta\cdot 180/\pi\).
5) Related circle quantities (often asked with arcs)
-
Circumference:
\[
C = 2\pi r.
\]
-
Chord length (endpoints of the arc):
\[
c = 2r\sin\left(\frac{\theta}{2}\right).
\]
-
Sector area (the “pizza slice”):
\[
A_{\text{sector}}=\frac{1}{2}r^2\theta \quad (\theta\text{ in radians}).
\]
-
Circular segment area (sector minus triangle):
\[
A_{\text{segment}}=\frac{1}{2}r^2\left(\theta-\sin\theta\right).
\]
Example
Given \(r=5\) and \(\theta=90^\circ\):
\[
s=\frac{90}{360}\cdot 2\pi\cdot 5
=\frac{1}{4}\cdot 10\pi
=\frac{5\pi}{2}
\approx 7.85.
\]
