Formula Used
The main radian formula is:
θ = s / r
Here, θ is the central angle in radians. The value s is arc length. The value r is radius.
Other supported formulas are θ = 2s / d, θ = 2πs / C, θ = 2A / r², and θ = 2 sin⁻¹(c / 2r).
How to Use This Calculator
Select the method that matches your known values.
Enter only the values needed for that method.
Use the precision box to control decimal places.
Press Calculate to see radians, degrees, normalized angle, and related circle values.
Use CSV or PDF buttons to save the final result table.
Example Data Table
| Known Values |
Formula |
Radian Angle |
Degree Angle |
| Arc = 12, Radius = 6 |
θ = s / r |
2 |
114.591559° |
| Arc = 5, Diameter = 10 |
θ = 2s / d |
1 |
57.295780° |
| Sector area = 20, Radius = 4 |
θ = 2A / r² |
2.5 |
143.239449° |
| Chord = 8, Radius = 5 |
θ = 2 sin⁻¹(c / 2r) |
1.854590 |
106.260205° |
Understanding Radian Measure
A central angle starts at the center of a circle. Its sides meet the circle at two points. The curved distance between those points is the arc length. Radian measure links that curved distance with the radius. This makes radians useful in geometry, trigonometry, physics, and engineering.
Why Radians Matter
Radians are natural circle units. One radian means the arc length equals the radius. A full circle has 2π radians. A straight angle has π radians. A right angle has π/2 radians. These forms make many formulas shorter. They also work better with sine, cosine, and calculus rules.
Advanced Input Choices
This calculator supports several common cases. You can use arc length with radius. You can use diameter instead of radius. You can also use circumference when the full boundary is known. Sector area is useful when an area slice is measured. Chord length helps when only the straight line across the arc is available.
Interpreting the Output
The main output is the central angle in radians. The tool also converts it into degrees. The multiple of π helps with exact style answers. The normalized angle places the result inside one full rotation. This is helpful when an angle is larger than 2π or negative.
Practical Uses
Students can use this tool to check homework. Teachers can prepare clean examples. Designers can estimate circular parts. Technicians can review pulley, wheel, and gear movement. The CSV export helps spreadsheet records. The PDF export helps reports and class notes.
Accuracy Notes
Use the same unit for arc, radius, diameter, chord, and circumference. Do not mix inches with centimeters. The radian answer has no length unit. It is a ratio. Increase decimal precision when you need more exact values. For chord calculations, the chord must not exceed the diameter.
FAQs
What is a central angle?
A central angle is formed by two radii. Its vertex is the circle center. It intercepts an arc on the circle.
What is radian measure?
Radian measure compares arc length with radius. If arc length equals radius, the angle is one radian.
What formula finds radians from arc length?
Use θ = s / r. Divide arc length by radius. The result is the central angle in radians.
Can I use diameter instead of radius?
Yes. Since radius is half the diameter, use θ = 2s / d. Enter arc length and diameter.
How many radians are in a full circle?
A full circle has 2π radians. That equals about 6.283185 radians, or 360 degrees.
How do I convert degrees to radians?
Multiply degrees by π / 180. This calculator also supports minutes and seconds for finer angles.
Why is my chord input rejected?
A chord cannot be longer than the diameter. Check that chord length is not greater than twice the radius.
Does the radian answer have units?
No. Radians are ratios. Use matching length units before division, and the units cancel out.