Radian (rad) - Unit Information & Conversion
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What is a Radian?
The radian is the SI unit of angle, defined by the radius of a circle. Used in mathematics, physics, and engineering for trigonometry, calculus, and angular measurements.
History of the Radian
First used by Roger Cotes in 1714, formally named "radian" by James Thomson in 1873. Adopted as SI unit in 1960. Natural unit for mathematical analysis.
Quick Answer
What is a Radian? A radian (rad) is the SI unit of angle equal to the angle subtended by an arc length equal to the radius. π radians = 180°, 2π radians = 360° (full circle). Essential for calculus and physics. Use our angle converter to convert radians to degrees and other units instantly.
Key Facts: Radian
| Property | Value |
|---|---|
| Symbol | rad |
| Quantity | Angle |
| System | Metric/SI Derived |
| Derived from | Radian |
| Category | Angle |
| Standard Body | NIST / ISO |
Quick Comparison Table
| Radians | Degrees | Context | Convert Now |
|---|---|---|---|
| 0 | 0° | Zero angle | rad to ° → |
| π/6 ≈ 0.524 | 30° | Common angle | rad to ° → |
| π/4 ≈ 0.785 | 45° | Diagonal | rad to ° → |
| π/3 ≈ 1.047 | 60° | Equilateral triangle | rad to ° → |
| π/2 ≈ 1.571 | 90° | Right angle | rad to ° → |
| π ≈ 3.142 | 180° | Straight angle | rad to ° → |
| 2π ≈ 6.283 | 360° | Full circle | rad to ° → |
Definition
A radian is the angle subtended at the center of a circle by an arc equal in length to the radius. 1 radian ≈ 57.2958 degrees.
Key values:
- π radians = 180°
- 2π radians = 360°
- 1 radian = 180/π ≈ 57.296°
Why Radians Matter
The radian is unique because it's dimensionless:
- Definition: Arc length ÷ Radius = Angle in radians
- Dimensionless: Ratio of two lengths (meters ÷ meters = just a number)
- Natural for math: Calculus formulas work perfectly with radians (no messy constants)
- SI standard: The official unit for angles in physics and engineering
- Degrees are arbitrary: Based on Babylonian base-60 system (historical accident)
Common Uses
Mathematics: Required unit for calculus (derivatives, integrals of trig functions only work with radians). Physics: Angular velocity (rad/s), rotational motion, oscillations, wave propagation. Engineering: Computer graphics, robotics, signal processing, control systems. Programming: All major languages (Python, JavaScript, C++) use radians for sin(), cos(), tan().
Real-World Applications
Common Radian Values in Practice
Quick reference for frequently used angles:
| Radians | Degrees | Description | Applications |
|---|---|---|---|
| 0 | 0° | No angle | Reference point |
| π/6 | 30° | Common angle | Mechanical design |
| π/4 | 45° | Diagonal | Optimal projectile angle |
| π/3 | 60° | Equilateral triangle | Crystallography |
| π/2 | 90° | Right angle | Perpendicular (most common) |
| 2π/3 | 120° | Obtuse | Three-way symmetry |
| π | 180° | Straight angle | Half-turn |
| 3π/2 | 270° | Three-quarter turn | Less common |
| 2π | 360° | Full circle | Complete rotation |
Angular Velocity in Engineering
Where radians per second (rad/s) matters:
| System | Angular Velocity | Radians/Second | RPM |
|---|---|---|---|
| Earth's rotation | 1 day⁻¹ | 7.27 × 10⁻⁵ | 0.0004 |
| Bicycle pedal | 60-120 RPM | 6.3-12.6 | 60-120 |
| Car engine | 600-7,000 RPM | 63-733 | 600-7,000 |
| Hard drive | 7,200 RPM | 754 | 7,200 |
| Dental drill | 300,000 RPM | 31,416 | 300,000 |
Why Radians Are Essential in Physics
Radian-based formulas vs degree-based (messy!):
| Formula | Using Radians | Using Degrees |
|---|---|---|
| Arc length | L = rθ | L = rθ × (π/180) |
| sin derivative | d/dθ[sin(θ)] = cos(θ) | d/dθ[sin(θ)] = (π/180)cos(θ) |
| Small angle | sin(θ) ≈ θ | sin(θ) ≈ θ × (π/180) |
Radian Conversion Formulas
To Degree:
To Gradian:
To Arcminute:
To Arcsecond:
To Turn:
To Revolution:
To Quadrant:
To Gon:
To Mil:
Frequently Asked Questions
Formula: degrees = radians × (180/π) = radians × 57.2958 Examples:
- 1 rad = 57.3°
- π/2 rad = 90°
- π rad = 180°
- 2π rad = 360°
Convert Radian
Need to convert Radian to other angle units? Use our conversion tool.