Radians to Degrees Converter
Convert angles between radians and degrees instantly.
Common Angles
| Degrees | Radians | π fraction | sin | cos |
|---|---|---|---|---|
| 0° | 0 | 0 | 0 | 1 |
| 30° | 0.523599 | π/6 | 0.5 | 0.866 |
| 45° | 0.785398 | π/4 | 0.7071 | 0.7071 |
| 60° | 1.0472 | π/3 | 0.866 | 0.5 |
| 90° | 1.5708 | π/2 | 1 | 6.123e-17 |
| 120° | 2.0944 | 2π/3 | 0.866 | -0.5 |
| 135° | 2.35619 | 3π/4 | 0.7071 | -0.7071 |
| 150° | 2.61799 | 5π/6 | 0.5 | -0.866 |
| 180° | 3.14159 | π | 1.225e-16 | -1 |
| 270° | 4.71239 | 3π/2 | -1 | -1.837e-16 |
| 360° | 6.28319 | 2π | -2.449e-16 | 1 |
Click any row to load that angle.
All calculations run live in your browser using JavaScript Math functions. π = 3.141592653589793.
What Is the Radians to Degrees Converter?
Degrees and radians are two ways to measure angles. Degrees divide a full circle into 360 parts, a convention from ancient Babylonian astronomy. Radians relate the angle to arc length on the unit circle: one radian is the angle that subtends an arc equal in length to the radius. Radians are the natural unit for calculus and most scientific computation.
- ›Type in degrees to instantly see radians, gradians, turns, and trig values
- ›Type in radians to convert the other way, both fields update in real time
- ›Click any row in the Common Angles table to load that angle
- ›All six outputs (sin, cos, tan, π-fraction, gradians, turns) update live
Formula
Degrees → Radians
rad = deg × π/180
Radians → Degrees
deg = rad × 180/π
Other conversions:
- ›Gradians: gon = deg × 10/9, full circle = 400 gon
- ›Turns: turns = deg / 360, one turn = one full revolution
- ›Arc length: s = r × θ (θ in radians), fundamental circle formula
How to Use
- 1Enter an angle in the Degrees field, all outputs update immediately
- 2Or enter radians in the Radians field, the degrees field updates in real time
- 3Use the Common Angles table to quickly load standard angles (30°, 45°, 90°, etc.)
- 4Results show: radians, π-fraction notation, gradians, turns, sin, cos, and tan
- 5Click Clear to reset both fields and results
- 6Your last entered angle is saved and restored on next visit
Example Calculation
Convert 270° to other units:
Gradians: 270 × 10/9 = 300 gon
Turns: 270 / 360 = 0.75 turns
sin(270°) = −1
cos(270°) = 0
tan(270°) = ±∞ (undefined)
Convert 2 radians to degrees
2 × (180/π) = 360/π ≈ 114.5916°
This angle is in the second quadrant (between 90° and 180°)
sin(2 rad) ≈ 0.9093, cos(2 rad) ≈ −0.4161
Understanding Radians to Degrees Converter
Common Angle Reference
| Degrees | Radians | π fraction | sin | cos | tan |
|---|---|---|---|---|---|
| 0° | 0 | 0 | 0 | 1 | 0 |
| 30° | 0.5236 | π/6 | 0.5 | 0.8660 | 0.5774 |
| 45° | 0.7854 | π/4 | 0.7071 | 0.7071 | 1 |
| 60° | 1.0472 | π/3 | 0.8660 | 0.5 | 1.7321 |
| 90° | 1.5708 | π/2 | 1 | 0 | ±∞ |
| 180° | 3.1416 | π | 0 | −1 | 0 |
| 270° | 4.7124 | 3π/2 | −1 | 0 | ±∞ |
| 360° | 6.2832 | 2π | 0 | 1 | 0 |
Frequently Asked Questions
Why are there two ways to measure angles?
The two systems exist for different purposes. Degrees are convenient for navigation and everyday geometry. Radians are essential in advanced mathematics because they make calculus formulas clean.
- ›Degrees: based on 360° for a full circle (Babylonian base-60 system)
- ›Radians: based on the ratio arc-length/radius, dimensionless and natural
- ›In calculus: d/dx[sin x] = cos x only works in radians
- ›In physics formulas: ω = 2πf uses radians per second
What is one radian?
A radian is defined geometrically: if you mark an arc along a circle equal in length to the radius, the central angle formed is exactly 1 radian.
- ›1 radian ≈ 57.2958° (exact: 180°/π)
- ›A full circle = 2π radians ≈ 6.2832 rad
- ›A semicircle = π radians ≈ 3.14159 rad
- ›A right angle = π/2 radians ≈ 1.5708 rad
Which unit should I use in programming?
A very common programming mistake is passing degrees to a trig function that expects radians. Always check the documentation.
- ›JavaScript: Math.sin(Math.PI / 2) = 1 (correct, in radians)
- ›Convert: radians = degrees * Math.PI / 180
- ›Python math module: also uses radians (use math.radians() to convert)
- ›Only a few libraries (like Excel's DEGREES/RADIANS functions) offer both
What are gradians (gon)?
The gradian system makes right angles a round number (100 gon) which is convenient for surveying. Some European theodolites still use gradians today.
- ›400 gon = 360° = 2π rad (full circle)
- ›100 gon = 90° = π/2 rad (right angle)
- ›200 gon = 180° = π rad (straight angle)
- ›Conversion: gon = degrees × 10/9
How do I convert common angles mentally?
A few key conversions are worth memorising, especially if you work with trigonometry or calculus regularly.
- ›30° = π/6 ≈ 0.5236 rad
- ›45° = π/4 ≈ 0.7854 rad
- ›60° = π/3 ≈ 1.0472 rad
- ›90° = π/2 ≈ 1.5708 rad
- ›180° = π ≈ 3.14159 rad
- ›360° = 2π ≈ 6.28318 rad
Why is tan(90°) undefined?
Tangent is the ratio of sine to cosine. At 90° (π/2), the cosine is exactly zero, making the ratio undefined. The graph has a vertical asymptote there.
- ›tan(θ) = sin(θ) / cos(θ)
- ›cos(90°) = 0, so tan(90°) is division by zero
- ›The same happens at 270° (3π/2) and any odd multiple of 90°
- ›The calculator shows ±∞ to indicate the discontinuity
What is the arc length formula?
The arc length formula is the most fundamental reason radians exist. It is clean and dimensionless only in radians.
- ›s = rθ (θ in radians)
- ›In degrees: s = r × (θ° × π/180), requires the conversion factor
- ›Sector area = ½r²θ, also clean in radians
- ›Angular velocity ω = θ/t, measured in radians per second