Find radius from radians
Use arc length for curved distance. Use chord length for straight distance between arc endpoints.
Example Data Table
| Radians | Method | Known Length | Radius |
|---|---|---|---|
| 1.5708 | Arc length | 12 cm | 7.6394 cm |
| 2.0944 | Chord length | 10 cm | 5.7735 cm |
| 0.7854 | Arc length | 20 in | 25.4648 in |
| 3.1416 | Chord length | 18 ft | 9.0000 ft |
Formula Used
In this equation, r is radius, s is arc length, and θ is the angle in radians.
Here, c is chord length. The chord method needs an angle above zero and below 2π radians. The calculator converts length units before applying either formula.
How to Use This Calculator
- Enter the central angle in radians.
- Select arc length or chord length as the known measurement.
- Enter the matching positive length value.
- Select the known length unit and desired result unit.
- Choose decimal places for display precision.
- Select Calculate Radius to view the result above the form.
- Use the CSV button or print option after calculation.
Radians, Length, and Radius
Circle Basics
A radius is the distance from a circle's center to its edge. Radians describe an angle through the relationship between an arc and its radius. A complete turn contains 2π radians. A half turn contains π radians. This calculator finds a radius when you know an angle in radians and either an arc length or chord length. It supports common length units. It also provides diameter, circumference, and degree equivalents. These results help check drawings, estimates, and measured circular parts.
Arc Length Method
The arc length method is direct. Arc length equals radius multiplied by the angle in radians. Rearranging the relationship gives radius equals arc length divided by radians. An arc length of fifteen centimeters with an angle of 1.5 radians gives a radius of ten centimeters. This method helps with sectors, bends, curved paths, and circular components. The angle must be entered in radians. Convert degree values before using this formula. Degree inputs cannot be substituted without conversion.
Chord Length Method
The chord method uses another relationship. A chord is a straight line joining two points on a circle. The chord formula is chord equals two times radius times sine of half the angle. Rearranging it finds radius from the chord and angle. This option helps when a curved edge is difficult to measure. It works for angles above zero and below two π radians. Near a complete turn, chord results become unstable. The chord approaches zero there.
Units and Precision
Unit choice matters. Enter the known length in the unit you measured. Then select the unit for the answer. The calculator converts values before showing the result. This avoids manual multiplication and reduces mistakes. You can work with millimeters, centimeters, meters, inches, feet, or yards. Keep notes consistent. A small unit mismatch can create a major fabrication error. Confirm the output unit before recording or sharing your result.
Reliable Results
Use a sensible precision setting. Two decimal places work for many classroom examples. Three or four places can help with machining, layout, and digital modeling. More decimals do not improve a rough measurement. They only show extra digits. Round the final figure according to project tolerance. Keep the unrounded value during intermediate work when several calculations depend on the radius. This practice prevents compounding rounding errors.
Practical Checks
Radius calculations appear in engineering, design, building, surveying, and education. They help define pipe bends, wheels, gears, arc layouts, and curved interfaces. Review each answer before using it. Radius must be positive. Diameter must be twice the radius. Circumference must equal two π times the radius. Recheck the original angle, length, unit, and method. Compare the result with drawing dimensions and expected scale. For arc entries, multiply radius by radians to check the original arc length. For chord entries, apply the sine relationship to confirm the stated chord. Small checks detect typing errors before cutting materials or issuing plans on site. Accurate radius calculations support confident geometry decisions every day.
FAQs
1. Can radians alone determine a radius?
No. Radians describe an angle, not a size. You also need an arc length, a chord length, or another matching circle measurement to calculate radius.
2. Which measurement method should I choose?
Choose arc length when you know the curved distance. Choose chord length when you know the straight distance between the two arc endpoints.
3. Why must the angle be in radians?
The arc length formula uses radians by definition. Degree values need conversion first, or the calculated radius will be incorrect.
4. How do I convert degrees to radians?
Multiply degrees by π and divide by 180. For example, 90 degrees equals π/2 radians, or about 1.5708 radians.
5. Can I use inches and receive meters?
Yes. Select inches for the known length and meters for the result. The calculator converts the measurement automatically before calculating radius.
6. What happens when radians are very small?
A very small angle can create a large radius when the known length stays fixed. Use enough decimal precision to avoid misleading rounding.
7. Why is chord input limited below 2π?
At 2π radians, the chord becomes zero. Values near that limit make the denominator extremely small and can produce unstable results.
8. Does the calculator show sector area?
Yes. It shows the area of the sector formed by the radius and angle. The output uses square units matching the selected result unit.
9. How can I verify an arc-length result?
Multiply the calculated radius by the entered radians. The result should equal the original arc length after unit conversion and rounding.
10. Is a radius always positive?
For physical circles, yes. This calculator requires positive radians and length inputs, then returns a positive radius measurement.
11. Can I save the result?
After a successful calculation, download a CSV copy or use the print option to save the displayed result as a PDF.