Free Arc Length Calculator

The Arc Length Calculator: A Versatile Tool for Circles

This free online arc length calculator handles not only the length of an arc and the area of the corresponding sector but also acts as a chord length calculator. Whether you are dealing with a simple homework problem or a real‑world design, the tool can determine any missing parameter when you supply two known values.

Arc Length and Sector Area – The Basics

For any circle, the arc length $L$ subtended by a central angle $\theta$ (measured in radians) is directly proportional to that angle. Because a full circle ($2\pi$ radians) corresponds to the full circumference $2\pi r$, we have:

$$\frac{L}{\theta} = \frac{2\pi r}{2\pi} = r \quad \Rightarrow \quad L = r\,\theta$$

The sector area $A$ follows the same logic. The whole circle area is $\pi r^{2}$, so:

$$\frac{A}{\theta} = \frac{\pi r^{2}}{2\pi} = \frac{r^{2}}{2} \quad \Rightarrow \quad A = \frac{r^{2}\,\theta}{2}$$

These two formulas are the foundation of all arc length and sector area calculations.

Working with Different Known Values

The tool can handle a variety of input combinations. If you have the radius and the central angle (in radians or degrees—the tool will convert degrees to radians automatically), apply the formulas above directly.

If the radius is unknown but the sector area and angle are given:

$$r = \sqrt{\frac{2A}{\theta}}, \quad \text{then } L = r\theta.$$

If the chord length $c$ and the central angle are known:

$$r = \frac{c}{2\sin(\theta/2)}, \quad L = r\theta.$$

Conversely, when the angle is missing but the radius and chord length are available:

$$\theta = 2\arcsin\left(\frac{c}{2r}\right), \quad L = r\theta.$$

Or, using the sector area and radius:

$$\theta = \frac{2A}{r^{2}}, \quad L = r\theta.$$

Step-by‑Step Example

Pick a practical scenario. Let the circle have a radius of 15 cm and a central angle of $45^{\circ}$ (i.e., $\pi/4$ radians).

• Arc length:
  $L = 15 \times \frac{\pi}{4} \approx 11.78\ \text{cm}$

• Sector area:
  $A = \frac{15^{2} \times \pi/4}{2} \approx 88.36\ \text{cm}^{2}$

The same procedure works for any radius or angle. If you only have the diameter, the calculator can accept that as well.

Why Radians Simplify Everything

Although you can express the central angle in any unit (degrees, minutes, arcseconds), the formulas become clean only when the angle is in radians. In degrees, a conversion factor $\pi/180$ appears, making the calculation slightly longer. The arc length itself is a linear distance (centimeters, meters, inches, etc.), never an angular measure.

The free arc length calculator, which also serves as a sector area calculator and a chord length calculator, removes the manual conversion burden. Enter any two parameters—radius and angle, radius and chord length, area and angle, etc.—and it instantly produces the remaining values, including the central angle in both radians and degrees if needed.

This comprehensive instrument is ideal for students, engineers, and anyone working with circular arcs and sectors.