Convert Angle Units

Enter an angle value and select the units to convert between:

Conversion Formulas:
Degrees ↔ Radians: multiply by π/180 or 180/π
Degrees ↔ Gradians: multiply by 10/9 or 9/10

Angle Value

From Unit

To Unit

Decimal Precision

Show Common Angles Table

Understanding Angle Measurement Systems

Angles can be measured in different units depending on the context and application. Each system has its own advantages and is used in specific fields of mathematics, science, and engineering.

Angle Measurement Systems

📐 Degrees (°)

Most common system worldwide
Divides circle into 360 equal parts
Based on ancient Babylonian mathematics
Easy for mental calculations
Used in everyday navigation

🔢 Radians (rad)

SI unit for angle measurement
Based on circle's radius property
One radian = arc length = radius
Natural for calculus applications
Essential in physics and engineering

📏 Gradians (grad)

Also called gon or gradian
Divides right angle into 100 parts
Decimal system simplifies calculations
Used in some surveying applications
Less common but mathematically convenient

Common Angle Conversions

Degrees	Radians	Gradians	Description
0°	0 rad	0 grad	Zero angle
30°	π/6 ≈ 0.5236 rad	33.333 grad	1/12 of circle
45°	π/4 ≈ 0.7854 rad	50 grad	1/8 of circle
60°	π/3 ≈ 1.0472 rad	66.667 grad	1/6 of circle
90°	π/2 ≈ 1.5708 rad	100 grad	Right angle
120°	2π/3 ≈ 2.0944 rad	133.333 grad	1/3 of circle
135°	3π/4 ≈ 2.3562 rad	150 grad	3/8 of circle
150°	5π/6 ≈ 2.6180 rad	166.667 grad	5/12 of circle
180°	π ≈ 3.1416 rad	200 grad	Straight angle
270°	3π/2 ≈ 4.7124 rad	300 grad	3/4 of circle
360°	2π ≈ 6.2832 rad	400 grad	Full circle

Trigonometric Functions for Common Angles

Angle	sin(θ)	cos(θ)	tan(θ)
0°	0	1	0
30°	1/2 = 0.5	√3/2 ≈ 0.866	√3/3 ≈ 0.577
45°	√2/2 ≈ 0.707	√2/2 ≈ 0.707	1
60°	√3/2 ≈ 0.866	1/2 = 0.5	√3 ≈ 1.732
90°	1	0	∞ (undefined)

Applications in Different Fields

📐 Mathematics

Trigonometry calculations
Triangle angle measurements
Complex number theory
Calculus applications
Geometric constructions

⚛️ Physics

Circular motion calculations
Wave phase angles
Vector component analysis
Rotational dynamics
Electromagnetic field theory

🏗️ Engineering

Structural angle calculations
Navigation and surveying
Mechanical design angles
Electrical phase angles
Robotics and automation

Navigation and Surveying

🧭 Compass Navigation

Bearing angle measurements
Magnetic declination corrections
Course plotting calculations
Position determination
Direction finding

📏 Surveying

Land surveying angles
Theodolite measurements
Property boundary calculations
Elevation angle determination
Construction layout

✈️ Aviation

Flight path calculations
Heading and bearing
Approach angle measurements
Runway slope calculations
Navigation system angles

Conversion Examples

🔄 Degrees to Radians

90° × (π/180) = π/2 radians
180° × (π/180) = π radians
360° × (π/180) = 2π radians
Formula: radians = degrees × π/180

🔄 Radians to Degrees

π radians × (180/π) = 180°
π/2 radians × (180/π) = 90°
2π radians × (180/π) = 360°
Formula: degrees = radians × 180/π

🔄 Degrees to Gradians

90° × (10/9) = 100 gradians
180° × (10/9) = 200 gradians
360° × (10/9) = 400 gradians
Formula: gradians = degrees × 10/9

Why Different Angle Systems Exist

📜 Historical Development

Degrees: Ancient Babylonian (360°)
Radians: 18th century mathematicians
Gradians: 20th century surveying
Each system evolved for specific needs
Modern usage depends on context

🔬 Scientific Applications

Radians: Natural for circular motion
Degrees: Human-friendly measurements
Gradians: Decimal calculation advantages
Context determines best choice
Some fields use multiple systems

🛠️ Practical Considerations

Degrees: Navigation, construction
Radians: Physics, engineering calculations
Gradians: Specialized surveying applications
Conversion tools essential for interdisciplinary work

📐 Angle Tip: Remember that π radians = 180 degrees. This fundamental relationship allows conversion between the two most commonly used angle measurement systems. Use radians for mathematical calculations and degrees for practical measurements.

Angle Converter