Arc Length Calculator

How Arc Length Calculator Works?

The Arc Length Calculator finds the distance along a circular curve from the circle's radius and central angle. The central angle must be measured at the center of the circle, and the resulting arc length follows the circle's radius and selected length unit.

In this formula, $s$ is the arc length, $r$ is the radius, and $\theta$ is the central angle expressed in radians. When the angle is entered in another supported format, the equivalent calculation is:

For example, with a radius of 5 m and a central angle of 90 degrees, the arc represents one quarter of the circle. The result is $s=5\times \frac{\pi }{2}\approx 7.854$ m. The calculator accepts radius values from 0.000001 to 1,000,000,000,000 and central angles from 0.000001 to 100,000, using degrees, radians, gradians, or turns for the angle.

Practical Use Cases

Students: Students can use the calculator to check geometry homework involving sectors, circular arcs, and central angles. Entering a known radius and angle provides a quick way to verify a hand calculation using the arc-length formula.

Engineers: Engineers can estimate the curved distance along pipes, gears, tracks, circular paths, and other components when a design specifies a radius and angular span.

Designers and architects: Designers can calculate the length of curved edges, rounded walls, circular features, and decorative arcs for planning materials and dimensions.

Surveyors and mapping professionals: Surveyors can determine the distance along a circular route or curved boundary when the radius and subtended angle are known.

Manufacturing and fabrication professionals: Fabricators can use arc length to plan bends, cut curved materials, and estimate the developed length of circular sections.

Accuracy and Limitations

The calculator applies the ideal-circle relationship $s=r\theta$ and reports a numerical result based on the entered radius and central angle. Its accuracy is limited by the precision of the supplied inputs and the calculator's numeric rounding. It assumes a constant radius and a true central angle; it does not calculate the length of an ellipse, an irregular curve, or a path whose radius changes. Radius values must be between 0.000001 and 1,000,000,000,000, and central angles must be between 0.000001 and 100,000 in the selected angle format. Measurement uncertainty, physical material thickness, and real-world construction tolerances are not included.

Quick Examples