Angle Converter

Convert angles between degrees, radians, gradians, mils, turns, arcminutes, and arcseconds with our comprehensive angle conversion calculator. Get instant, accurate results with trigonometric values and detailed explanations.

Angle Converter
7 Units

Convert between degrees, radians, gradians, mils, and other angle measurements

Angle Value

From Unit

Common Angle References

Standard angles for reference and understanding

Basic

Right Angle

90°

Quarter turn, perpendicular lines

1.5708 rad | 100.00 gon

Straight Angle

180°

Half turn, straight line

3.1416 rad | 200.00 gon

Full Circle

360°

Complete rotation

6.2832 rad | 400.00 gon

Acute Angle

45°

Less than 90°

0.7854 rad | 50.00 gon

Obtuse Angle

135°

Greater than 90°, less than 180°

2.3562 rad | 150.00 gon

Geometry

Pentagon Interior

108°

Interior angle of regular pentagon

1.8850 rad | 120.00 gon

Hexagon Interior

120°

Interior angle of regular hexagon

2.0944 rad | 133.33 gon

Octagon Interior

135°

Interior angle of regular octagon

2.3562 rad | 150.00 gon

Special

Golden Angle

137.5°

Related to golden ratio in nature

2.3998 rad | 152.78 gon

Navigation

Compass Point

22.5°

1/16 of full circle

0.3927 rad | 25.00 gon

Understanding Angle Measurement Units

Angles are fundamental measurements in mathematics, physics, engineering, and navigation. Different angle units have evolved for specific applications, each offering unique advantages for particular calculations and contexts. Understanding these units and their relationships is essential for accurate mathematical work and practical applications.

Degrees (°)

Degrees are the most familiar angle unit for most people. The degree system divides a full circle into 360 equal parts, making it intuitive for everyday use and geometric calculations.

• Full circle: 360°
• Right angle: 90°
• Straight angle: 180°
• Common in geometry and navigation
• Easy to visualize and understand

Radians (rad)

Radians are the standard unit of angular measurement in mathematics and physics. One radian is the angle subtended by an arc length equal to the radius of the circle.

• Full circle: 2π radians ≈ 6.283
• Right angle: π/2 radians ≈ 1.571
• Straight angle: π radians ≈ 3.142
• Natural unit for calculus
• Simplifies trigonometric formulas

Gradians (gon)

Gradians (also called gons) divide a full circle into 400 equal parts. This decimal-based system was designed to work well with metric measurements and decimal calculations.

• Full circle: 400 gradians
• Right angle: 100 gradians
• Straight angle: 200 gradians
• Used in surveying and engineering
• Decimal-friendly calculations

Mils (mil)

Mils are primarily used in military and artillery applications. The NATO standard defines 6400 mils in a full circle, making range and bearing calculations more practical for field use.

• Full circle: 6400 mils
• Right angle: 1600 mils
• Straight angle: 3200 mils
• Used in military navigation
• Practical for artillery calculations

Turns

Turns represent complete rotations, where 1 turn equals a full 360° rotation. This unit is intuitive for describing rotational motion and frequency.

• Full circle: 1 turn
• Half turn: 0.5 turns
• Quarter turn: 0.25 turns

Arcminutes (′)

Arcminutes are 1/60 of a degree, used for precise angular measurements in astronomy, navigation, and optics.

• 60 arcminutes = 1 degree
• Used in celestial navigation
• Precision optical measurements

Arcseconds (″)

Arcseconds are 1/3600 of a degree, providing extremely precise angular measurements for scientific and astronomical applications.

• 3600 arcseconds = 1 degree
• 60 arcseconds = 1 arcminute
• High-precision astronomy

Angle Conversion Formulas and Mathematics

Converting between different angle units requires understanding the mathematical relationships between them. All conversions are based on the fundamental relationship that these units represent different ways of dividing a complete circle.

Primary Conversions

Degrees ↔ Radians:
rad = deg × (π/180)
deg = rad × (180/π)

Degrees ↔ Gradians:
gon = deg × (400/360)
deg = gon × (360/400)

Degrees ↔ Mils:
mil = deg × (6400/360)
deg = mil × (360/6400)

Precision Conversions

Degrees ↔ Turns:
turn = deg / 360
deg = turn × 360

Degrees ↔ Arcminutes:
arcmin = deg × 60
deg = arcmin / 60

Degrees ↔ Arcseconds:
arcsec = deg × 3600
deg = arcsec / 3600

Understanding the Radian

The radian is defined as the angle subtended by an arc whose length equals the radius of the circle. This natural definition makes radians particularly useful in calculus and physics:

Mathematical Advantages:

• Simplifies derivatives: d/dx(sin x) = cos x (when x is in radians)
• Arc length formula: s = rθ (where θ is in radians)
• Sector area formula: A = ½r²θ
• Natural unit for oscillatory motion

Common Radian Values:

• π/6 rad = 30°
• π/4 rad = 45°
• π/3 rad = 60°
• π/2 rad = 90°
• π rad = 180°
• 2π rad = 360°

Practical Applications of Angle Conversion

Different angle units are preferred in various fields due to their specific advantages and historical usage. Understanding when and why to use each unit is crucial for accurate calculations and effective communication in professional contexts.

🔬 Physics & Engineering

Radians dominate in physics and engineering calculations due to their mathematical properties:

• Rotational dynamics calculations
• Wave mechanics and oscillations
• Control systems engineering
• Signal processing and Fourier analysis
• Mechanical engineering design

🗺️ Navigation & Surveying

Different units serve specific navigation and surveying needs:

• Degrees: GPS coordinates, compass bearings
• Gradians: European surveying standards
• Mils: Military navigation and artillery
• Arcminutes/seconds: Precise positioning
• Marine and aviation navigation

🌟 Astronomy & Optics

High-precision angle measurements are essential in astronomical work:

• Star position measurements
• Telescope pointing accuracy
• Planetary motion calculations
• Satellite tracking systems
• Optical instrument calibration

🏗️ Architecture & Construction

Building design and construction rely heavily on angle measurements:

• Roof pitch calculations
• Structural joint angles
• Site surveying and layout
• Solar panel positioning
• Architectural design software

🎮 Computer Graphics & Gaming

Digital applications use various angle representations:

• 3D rotation transformations
• Animation keyframe interpolation
• Camera movement calculations
• Physics simulation engines
• User interface design

🏭 Manufacturing & Machining

Precision manufacturing requires accurate angle conversions:

• CNC machine programming
• Tool angle specifications
• Quality control measurements
• Robotic arm positioning
• Assembly line automation

Trigonometry and Special Angles

Certain angles appear frequently in mathematics and have special properties that make calculations easier. Understanding these special angles and their trigonometric values is essential for efficient problem-solving in mathematics, physics, and engineering.

Degrees	Radians	Gradians	sin	cos	tan
0°	0	0	0	1	0
30°	π/6	33.33	1/2	√3/2	√3/3
45°	π/4	50	√2/2	√2/2	1
60°	π/3	66.67	√3/2	1/2	√3
90°	π/2	100	1	0	∞
180°	π	200	0	-1	0
270°	3π/2	300	-1	0	∞
360°	2π	400	0	1	0

Unit Circle Properties

The unit circle (radius = 1) provides a visual representation of trigonometric functions and angle relationships:

• Coordinates (cos θ, sin θ) for any angle θ
• Angles measured counterclockwise from positive x-axis
• Quadrant I: both sin and cos positive
• Quadrant II: sin positive, cos negative
• Quadrant III: both sin and cos negative
• Quadrant IV: sin negative, cos positive

Angle Identities

Important trigonometric identities that remain consistent across angle units:

• sin²θ + cos²θ = 1 (Pythagorean identity)
• tan θ = sin θ / cos θ
• sin(θ + 2π) = sin θ (periodicity)
• cos(θ + 2π) = cos θ (periodicity)
• sin(-θ) = -sin θ (odd function)
• cos(-θ) = cos θ (even function)

Best Practices for Angle Conversion

Conversion Accuracy Tips

Maintain Precision

Use high-precision values for π (3.14159265359) in calculations. Avoid rounding intermediate results to prevent cumulative errors.

Choose Appropriate Units

Select the most suitable unit for your application: radians for calculus, degrees for geometry, gradians for surveying.

Verify Results

Cross-check conversions using known reference angles (30°, 45°, 60°, 90°) to ensure accuracy.

Common Pitfalls to Avoid

Calculator Mode Confusion

Always verify your calculator is in the correct mode (DEG/RAD/GRAD) before performing trigonometric calculations.

Unit Inconsistency

Ensure all angles in a calculation use the same unit system. Mixing units is a common source of errors.

Angle Normalization

Remember that angles can be normalized (e.g., 450° = 90°) and consider coterminal angles in your calculations.

Professional Application Guidelines

Scientific Computing

Use radians for all mathematical computations, especially in programming and scientific software. Most mathematical libraries expect radian input.

Engineering Documentation

Clearly specify angle units in all technical documentation. Include conversion factors when working with international teams.

Educational Context

Introduce degrees first for intuitive understanding, then progress to radians for advanced mathematics and physics applications.