What Is Arc Length?
The arc length is the distance measured along the curved edge of a circle between two points. When the central angle subtended by the arc is expressed in radians, the relationship is wonderfully simple: \(s = r \times \theta\). This makes radians the natural unit for circular measurement — the angle directly scales the radius into a length.
How to Use This Calculator
Enter the circle's radius (\(r\)) and the central angle (\(\theta\)) in radians. The calculator instantly returns the arc length in the same units as your radius. It also gives the chord length — the straight-line distance between the two endpoints of the arc — for reference. If your angle is in degrees, convert it first by multiplying by \(\pi/180\).
The Formula Explained
A full circle is \(2\pi\) radians and has a circumference of \(2\pi r\). An arc covering a fraction \(\theta/(2\pi)\) of the circle therefore has length $$\left(\frac{\theta}{2\pi}\right) \cdot 2\pi r = r\theta.$$ The chord uses the isosceles triangle formed by two radii and the chord: $$c = 2r \cdot \sin\!\left(\frac{\theta}{2}\right).$$
Worked Example
Suppose \(r = 5\) and \(\theta = 1.5708\) radians (90°). Then $$s = 5 \times 1.5708 = 7.854 \text{ units}.$$ The chord is $$c = 2 \times 5 \times \sin(0.7854) = 10 \times 0.7071 = 7.071 \text{ units}.$$ As expected, the curved arc is slightly longer than the straight chord.
FAQ
Do I have to use radians? Yes — the \(s = r\theta\) formula only works with radians. Convert degrees with \(\theta = \text{degrees} \times \pi/180\).
What units does the answer use? Arc length is in the same units as the radius (cm, m, inches, etc.).
Why show the chord too? Many design and engineering tasks need both the curved distance and the straight-line span across the arc.
