Calculate arc length from radius and central angle in degrees or radians. Also find radius, central angle, sector area, and converted angle values in one tool.
Assumptions & Limitations
Use the Arc Length Calculator to calculate arc length from radius and central angle in either degrees or radians. The tool also reverse-calculates radius or central angle from known values, automatically outputs the corresponding sector area, and provides continuous degree-to-radian conversions. All linear measurements share a single, user-defined unit, while a built-in logic check flags multi-turn arcs when angles exceed a standard $360^\circ$ or $2\pi$ rotation.
What the arc length calculator solves
Depending on your known variables, the calculator adjusts its core logic to solve for the missing dimension. The table below outlines the five operational modes, the required inputs for each, and the primary and secondary outputs generated.
| Calculation method | Inputs used | Main result | Supporting outputs |
|---|---|---|---|
| Find Arc Length (Angle in Degrees) | Radius, central angle in degrees | Arc length | Central angle in radians, sector area |
| Find Arc Length (Angle in Radians) | Radius, central angle in radians | Arc length | Central angle in degrees, sector area |
| Find Radius (Angle in Degrees) | Arc length, central angle in degrees | Radius | Central angle in radians, sector area |
| Find Radius (Angle in Radians) | Arc length, central angle in radians | Radius | Central angle in degrees, sector area |
| Find Central Angle | Arc length, radius | Central angle in degrees and radians | Sector area |
Arc length formula
The standard mathematical approach calculates arc length using radians. However, practical applications often use degrees, requiring a conversion factor within the formula. The formulas below power the calculator’s exact logic.
| Use case | Formula |
|---|---|
| Arc length with radians | $$s = r\theta$$ |
| Arc length with degrees | $$s = 2\pi r \times \left(\frac{\theta}{360}\right)$$ |
| Radius from arc length and radians | $$r = \frac{s}{\theta}$$ |
| Radius from arc length and degrees | $$r = \frac{s}{\theta \times \left(\frac{\pi}{180}\right)}$$ |
| Central angle in radians | $$\theta = \frac{s}{r}$$ |
| Central angle in degrees | $$\theta = \left(\frac{s}{r}\right) \times \left(\frac{180}{\pi}\right)$$ |
| Sector area with radians | $$A = \frac{1}{2} r^2 \theta$$ |
| Sector area with degrees | $$A = \pi r^2 \times \left(\frac{\theta}{360}\right)$$ |
Variable definitions:
- $s$ = arc length
- $r$ = radius
- $\theta$ = central angle
- $A$ = sector area
How to use the arc length calculator
The interface is designed for immediate calculation without manual unit conversions. Follow the sequence below to get your results:
- Choose the calculation method.
- Enter the two required values for that mode.
- Select the linear unit for radius and arc length.
- Read the main result and the related outputs.
- Check the warning if the angle is greater than a full circle.
If you are unsure which mode to select, match your known values to the correct tool function:
| If you know | Select mode |
|---|---|
| Radius and angle in degrees | Find Arc Length (Angle in Degrees) |
| Radius and angle in radians | Find Arc Length (Angle in Radians) |
| Arc length and angle in degrees | Find Radius (Angle in Degrees) |
| Arc length and angle in radians | Find Radius (Angle in Radians) |
| Arc length and radius | Find Central Angle |
Degrees to radians and radians to degrees
Because central angles are measured in both standard systems, the calculator engine automatically converts your input to provide a complete data set. The conversions rely on the constant ratio between a full circle’s degrees ($360^\circ$) and its radian equivalent ($2\pi$).
| Conversion | Formula |
|---|---|
| Degrees to radians | $$\text{rad} = \text{deg} \times \left(\frac{\pi}{180}\right)$$ |
| Radians to degrees | $$\text{deg} = \text{rad} \times \left(\frac{180}{\pi}\right)$$ |
Quick-reference angle table:
| Degrees | Radians |
|---|---|
| $30^\circ$ | $$\frac{\pi}{6}$$ |
| $45^\circ$ | $$\frac{\pi}{4}$$ |
| $60^\circ$ | $$\frac{\pi}{3}$$ |
| $90^\circ$ | $$\frac{\pi}{2}$$ |
| $180^\circ$ | $$\pi$$ |
| $270^\circ$ | $$\frac{3\pi}{2}$$ |
| $360^\circ$ | $$2\pi$$ |
Arc length, radius, angle, and sector area relationships
Circle sectors are defined by rigidly proportional relationships. Changing one dimension forces a recalculation of the others. The calculator maps the relationships below so you can cross-check outputs.
| Known values | Can solve |
|---|---|
| Radius + central angle | Arc length, sector area |
| Arc length + central angle | Radius, sector area |
| Arc length + radius | Central angle, sector area |
| Radius + central angle in degrees | Central angle in radians |
| Radius + central angle in radians | Central angle in degrees |
Crucially, the calculator binds the arc length and radius to the exact same linear unit. Consequently, the resulting sector area is always expressed in the squared version of that selected unit (e.g., if radius is in meters, area is in square meters). The angle remains an independent rotational metric.
Worked examples for each calculator mode
To verify the tool’s accuracy or understand the underlying math, compare your manual calculations against the exact scenarios below. Each example demonstrates the formula pathway the tool executes based on the selected mode.
| Mode | Inputs | Formula path | Solved outputs |
|---|---|---|---|
| Arc length with degrees | $r = 5$, $\theta = 60^\circ$ | Convert angle to radians, then $s = r\theta$ | $s = 5.235987$, $\text{rad} = 1.047197$, $A = 13.089969$ |
| Arc length with radians | $r = 5$, $\theta = 1.047197$ | Use $s = r\theta$ directly | $s = 5.235987$, $\text{deg} = 60^\circ$, $A = 13.089969$ |
| Radius with degrees | $s = 5.235987$, $\theta = 60^\circ$ | Convert degrees to radians, then $r = \frac{s}{\theta}$ | $r = 5$, $\text{rad} = 1.047197$, $A = 13.089969$ |
| Radius with radians | $s = 5.235987$, $\theta = 1.047197$ | Use $r = \frac{s}{\theta}$ | $r = 5$, $\text{deg} = 60^\circ$, $A = 13.089969$ |
| Central angle | $s = 5.235987$, $r = 5$ | Use $\theta = \frac{s}{r}$, then convert | $\text{rad} = 1.047197$, $\text{deg} = 60^\circ$, $A = 13.089969$ |
When the result represents a multi-turn arc
A standard circle sector exists within a single rotation. However, mathematical formulas do not inherently cap at one circle. The calculator actively monitors the input angle and triggers a warning if the value exceeds $2\pi$ radians or $360^\circ$.
| Angle size | Interpretation |
|---|---|
| Less than a full circle | Standard single-turn sector arc |
| Equal to a full circle | Full circumference |
| Greater than a full circle | Multi-turn arc spanning more than one rotation |
The calculations remain mathematically precise for larger inputs, but the physical interpretation shifts from a simple pie slice to a continuous path overlapping itself across multiple rotations.
Input rules and calculator limits
To ensure mathematical validity and prevent broken outputs, the tool enforces strict input parameters before returning a result.
| Rule | Meaning |
|---|---|
| Both required inputs must be entered | The tool calculates only when the current mode has both values |
| Values must be numeric | Invalid text is rejected |
| Values must be greater than zero | Zero and negative values are not accepted |
| Linear unit applies to all linear values | Radius and arc length use the selected unit |
| Area is shown in squared units | Example: $\text{cm}^2$, $\text{m}^2$, $\text{in}^2$, $\text{ft}^2$ |
Choose the right arc length formula
Depending on the geometric values you need to find, use the guide below to select the correct formula and approach. The structure aligns directly with the calculator’s primary solving modes.
| What you want to do | Recommended formula or approach |
|---|---|
| Use an arc length calculator | Enter radius and angle to find $s = r\theta$ or $s = 2\pi r \times \left(\frac{\theta}{360}\right)$ |
| Find the standard arc length formula | Use $s = r\theta$ where $\theta$ is in radians |
| Find arc length with radius and angle | Multiply radius by the angle in radians |
| Calculate arc length in radians | $s = r\theta$ |
| Calculate arc length in degrees | $s = 2\pi r \times \left(\frac{\theta}{360}\right)$ |
| Find radius from arc length | $r = \frac{s}{\theta}$ |
| Use a central angle calculator | Divide arc length by radius ($\theta = \frac{s}{r}$) |
| Get sector area from radius and angle | Apply $A = \frac{1}{2} r^2 \theta$ using your known radius and angle |
| Convert degrees to radians for arc length | Multiply degrees by $\frac{\pi}{180}$ |
Related circle values the calculator does and does not solve
The calculator focuses exclusively on sector and arc relationships. It does not process full-circle inputs like diameter or straight-line measurements like chord length.
| Value | Directly calculated by the tool? |
|---|---|
| Arc length | Yes |
| Radius | Yes |
| Central angle | Yes |
| Sector area | Yes |
| Degree/radian conversion | Yes |
| Diameter | No direct mode |
| Chord length | No direct mode |
| Circumference | No direct output |
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