Angle Converter Calculator - Convert Degrees, Radians & More

Convert between degrees, radians, gradians, turns, and other angle units. Calculate trigonometric functions and reference angles.

Angle Converter
Convert between different angle units and calculate trigonometric values

Common Angle Presets

Conversion Results
All angle units and trigonometric values

Angle Units

Degrees
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Radians
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Gradians
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Turns
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Arc Minutes
--
Arc Seconds
--
Milliradians
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Trigonometric Functions

sin
--
cos
--
tan
--
cot
--
sec
--
csc
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Angle Information
Understanding angle units and conversions
Degrees (°)
Most common unit. Full circle = 360°. Used in everyday measurements and navigation.
Radians (rad)
Mathematical unit. Full circle = 2π rad. Used in calculus and physics.
Gradians (gon)
Metric unit. Full circle = 400 gon. Used in surveying and civil engineering.

Quick Reference

• 1 turn = 360° = 2π rad
• 1° = π/180 rad ≈ 0.01745 rad
• 1 rad = 180/π° ≈ 57.296°
• 1° = 60' = 3600"
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Quick Navigation

›Understanding Angles
›Angle Units Explained
›Conversion Formulas
›Trigonometric Functions
›Special Angles
›Practical Applications
›Coordinate Systems
›Common Conversions
›Tips & Tricks
›Frequently Asked Questions

Quick Tip: Angles are fundamental to mathematics, physics, engineering, and many practical applications. Understanding different angle units and their conversions is essential for working across various fields.

Understanding Angles and Their Measurement

Two rays sharing a common endpoint—the vertex—create what we call an angle, a geometric figure that's utterly fundamental to everything from navigating a ship to designing a bridge. The space between these rays can be measured in several ways, each serving different purposes across mathematics, physics, and engineering. You've likely encountered degrees in weather reports and navigation, but mathematicians and physicists gravitate toward radians for their elegant mathematical properties. The Clark University trigonometry tutorial offers deep insight into these measurement systems and their historical development. Our calculator bridges these different units of measurement, converting instantly between them while computing trigonometric values and revealing geometric relationships that might otherwise stay hidden.

📐 Geometric Angles

Fundamental to shapes, constructions, and spatial relationships in mathematics and design.

🔄 Rotational Motion

Essential for physics, engineering, and mechanics in describing circular and periodic motion.

🧭 Navigation

Critical for GPS, aviation, maritime navigation, and geographic positioning systems.

📊 Data & Graphics

Used in computer graphics, data visualization, and animation for transformations and rotations.

Angle Units Explained

Different angle units serve specific purposes across various fields. Understanding when and why to use each unit is crucial for effective communication and accurate calculations. Our converter supports all major angle units, from common degrees to specialized units like milliradians used in ballistics. Learn the conversion formulas and discover practical applications for each unit.

  • Degrees (°): The most familiar unit, dividing a circle into 360 parts. Originated from ancient Babylonian astronomy and their base-60 number system. Perfect for everyday use, navigation, and practical measurements.

  • Radians (rad): The mathematical standard where one radian equals the angle subtended by an arc equal to the radius. A full circle is 2π radians. Essential for calculus, physics, and engineering calculations.

  • Gradians (gon/grad): Divides a circle into 400 units, making a right angle exactly 100 gradians. Popular in surveying and some European countries for its metric-friendly decimal system.

  • Turns/Revolutions: One turn equals a full rotation (360° or 2π rad). Intuitive for describing rotational motion, used in physics and engineering for angular velocity and frequency.

  • Arc Minutes/Seconds: Subdivisions of degrees (1° = 60' = 3600"). Used for precise measurements in astronomy, GPS coordinates, surveying, and optics.

  • Milliradians (mrad): One thousandth of a radian (1 rad = 1000 mrad). Common in ballistics, rangefinding, and precision optics for small angle approximations.

🔄 Quick Conversion Reference

360°
= 2π rad = 400 gon = 1 turn
1 rad
≈ 57.296° = 63.662 gon
1°
= 60' = 3600" = π/180 rad

Angle Conversion Formulas

Mastering angle conversions requires understanding the mathematical relationships between different units. These formulas form the foundation for converting between various angle measurements and are essential for working across different fields. Whether you're solving trigonometric problems or working with different coordinate systems, these conversions are fundamental.

📏 From Degrees

To Radians:
radians = degrees × (π/180)
To Gradians:
gradians = degrees × (10/9)
To Turns:
turns = degrees / 360
To Arc Minutes:
arc minutes = degrees × 60

🔢 From Radians

To Degrees:
degrees = radians × (180/π)
To Gradians:
gradians = radians × (200/π)
To Turns:
turns = radians / (2π)
To Milliradians:
milliradians = radians × 1000

Trigonometric Functions and Angles

Trigonometric functions relate angles to ratios of sides in right triangles and are fundamental to mathematics, physics, and engineering. Our calculator computes all six primary trigonometric functions for any angle, helping you understand periodic behavior, wave functions, and circular motion. These functions are essential for special angles and have wide-ranging practical applications.

📊 Primary Trigonometric Functions

sin(θ)
Sine
Opposite / Hypotenuse
cos(θ)
Cosine
Adjacent / Hypotenuse
tan(θ)
Tangent
Opposite / Adjacent
csc(θ)
Cosecant
1 / sin(θ)
sec(θ)
Secant
1 / cos(θ)
cot(θ)
Cotangent
1 / tan(θ)

Special Angles and Their Properties

Certain angles appear frequently in mathematics and have exact trigonometric values that are worth memorizing. These special angles form the foundation of trigonometry and appear in countless applications from architecture to signal processing. Learning about their properties and relationships helps solve problems more efficiently and recognize patterns in periodic phenomena.

🌟 Common Special Angles

Anglesincostan
0°010
30°1/2√3/2√3/3
45°√2/2√2/21
60°√3/21/2√3
90°10∞

🔄 Angle Relationships

  • Reference Angles: Acute angle to x-axis
  • Complementary: Sum to 90° (π/2 rad)
  • Supplementary: Sum to 180° (π rad)
  • Coterminal: Differ by 360° (2π rad)
  • Quadrantal: 0°, 90°, 180°, 270°, 360°

Practical Applications of Angle Conversion

Picture a geologist measuring the tilt of rock strata, a pilot adjusting course mid-flight, or a game developer rotating a 3D model—each relies on precise angle conversions, though they might use entirely different units for the same rotation. Engineering blueprints speak in degrees while physics equations demand radians, creating a constant need for accurate translation between these systems. The Carleton College geoscience mathematics resource demonstrates how earth scientists employ trigonometry and angle measurement to calculate everything from hillside slopes to earthquake fault angles. Whether you're programming graphics transformations, surveying land, or analyzing wave functions, fluency in angle conversion transforms from a nice-to-have skill into an absolute necessity that separates competent work from excellence.

🚀 Engineering & Physics

• Rotational Motion: Angular velocity in rad/s
• Wave Functions: Phase angles in radians
• Mechanical Design: Gear ratios and cam profiles
• Signal Processing: Frequency and phase analysis

🧭 Navigation & Surveying

• GPS Coordinates: Degrees, minutes, seconds
• Compass Bearings: 0-360° from north
• Land Surveying: Gradians for slope calculations
• Aviation: Heading and course angles

💻 Computer Graphics

• 3D Rotations: Euler angles and quaternions
• Animation: Interpolation and tweening
• Game Development: Camera and object rotation
• Image Processing: Rotation transformations

Coordinate Systems and Angle Conventions

Different fields use various coordinate systems and angle conventions, making it essential to understand how angles are measured and interpreted in each context. From mathematical standards to navigation bearings, each system has its own reference point and direction of measurement. Our calculator helps you work seamlessly across these different conventions.

📐 Mathematical Standard

• Origin: Positive x-axis (3 o'clock)
• Direction: Counterclockwise positive
• Range: -∞ to +∞ (coterminal)
• Usage: Math, physics, engineering

🧭 Navigation/Compass

• Origin: North (12 o'clock)
• Direction: Clockwise positive
• Range: 0° to 360°
• Usage: Navigation, aviation, maritime

Common Angle Conversions Reference

Quick reference for frequently used angle conversions in various fields. These conversions appear regularly in calculations and are worth memorizing for efficiency. Whether you're working with trigonometric functions or converting between coordinate systems, these values form the foundation of angle mathematics.

📊 Quick Conversion Table

π/6 rad
= 30° = 33.33 gon
π/4 rad
= 45° = 50 gon
π/3 rad
= 60° = 66.67 gon
π/2 rad
= 90° = 100 gon

Tips and Tricks for Angle Conversions

Master angle conversions with these practical tips and mental math shortcuts. These techniques help you quickly estimate conversions and verify calculations, essential skills for exams, professional work, and everyday problem-solving.

✅ Quick Mental Math

• Degrees to radians: Divide by 60, multiply by π/3
• Radians to degrees: Multiply by 60, divide by π/3
• Right angle: 90° = π/2 rad = 100 gon
• Small angles: sin(θ) ≈ tan(θ) ≈ θ (in radians)

💡 Remember These

• π radians = 180°: Foundation for all conversions
• 1 radian ≈ 57.3°: Useful for estimates
• 1° ≈ 0.0175 rad: Quick small angle conversion
• Golden angle: 137.5° used in nature and design

Common Mistakes to Avoid

❌ Common Errors

• Calculator mode: Wrong DEG/RAD setting
• Unit confusion: Mixing degrees and radians
• Sign errors: Forgetting negative angles
• Precision loss: Rounding π too early

✅ Best Practices

• Check units: Always verify input/output units
• Use exact values: Keep π symbolic when possible
• Normalize angles: Reduce to 0-360° range
• Double-check: Verify with known values

Advanced Angle Concepts

Beyond basic conversions, angles play crucial roles in advanced mathematics and physics. Learning about concepts like solid angles (measured in steradians), hyperbolic angles, and complex number arguments opens doors to advanced applications in quantum mechanics, relativity, and signal processing. These concepts build upon the fundamental angle units and conversions covered by our calculator.

Modern applications continue to expand the importance of angle conversions. In robotics, precise angle control determines movement accuracy. In astronomy, tiny angular measurements (milliarcseconds) help measure stellar distances. In telecommunications, phase angles determine signal quality. Whether you're a student learning trigonometry or a professional working with complex rotational systems, Learning about angle conversions is essential for success in STEM fields.

Key Takeaways for Angle Conversion

Angles can be measured in multiple units, each with specific advantages. Degrees are intuitive for everyday use, radians simplify mathematical calculations, and specialized units like gradians and milliradians serve specific professional needs. Our calculator instantly converts between all major units.

Understanding trigonometric functions and their relationship to angles is fundamental for mathematics, physics, and engineering. Special angles like 30°, 45°, and 60° have exact values worth memorizing. Reference angles, complementary angles, and supplementary angles help solve complex trigonometric problems efficiently.

Different fields use various coordinate systems and angle conventions. Mathematical standards measure counterclockwise from the positive x-axis, while navigation uses clockwise from north. Understanding these conventions prevents errors when working across disciplines and ensures accurate communication.

Practical applications of angle conversion span from GPS navigation using degrees-minutes-seconds to computer graphics using radians for rotations. Master the mental math shortcuts and common conversions to work efficiently across different domains and solve real-world problems.

Frequently Asked Questions

most common angle units are degrees (360° in a circle), used in everyday measurements and navigation; radians (2π in a circle), used in mathematics and physics; gradians (400 in a circle), used in surveying and civil engineering; and turns (1 per circle), used in rotational measurements. Each unit has specific applications where it's most practical.
To convert degrees to radians, multiply by π/180 (approximately 0.01745). To convert radians to degrees, multiply by 180/π (approximately 57.296). For example, 90° = 90 × (π/180) = π/2 radians, and π radians = π × (180/π) = 180°. These conversions are essential for switching between mathematical and practical applications.
Arc minutes (') and arc seconds (") are subdivisions of degrees used for precise angular measurements. One degree equals 60 arc minutes, and one arc minute equals 60 arc seconds. They're commonly used in astronomy for celestial coordinates, navigation for GPS precision, surveying for land measurements, and optics for field of view specifications.
A reference angle is the acute angle (0° to 90°) formed between the terminal side of an angle and the x-axis. It's always positive and helps determine trigonometric values for any angle. Reference angles are vital for solving trigonometric equations, simplifying calculations, and Learning about periodic functions. They're the foundation for the unit circle concept.
Angles are classified as: acute (0° to 90°), right (exactly 90°), obtuse (90° to 180°), straight (exactly 180°), reflex (180° to 360°), and full rotation (360°). Special angles like 30°, 45°, 60°, and their multiples have exact trigonometric values. Learning about these classifications helps in geometry, trigonometry, and practical applications.
Complementary angles sum to 90° (like 30° and 60°), while supplementary angles sum to 180° (like 120° and 60°). These relationships are fundamental in geometry for solving angle problems, in trigonometry for identities (sin θ = cos(90°-θ)), and in practical applications like construction and design where perpendicular and straight lines are common.
Radians are the natural unit for angles in mathematics because they simplify many formulas. In calculus, derivatives and integrals of trigonometric functions are cleaner (d/dx sin(x) = cos(x) only works with radians). In physics, angular velocity (ω) and other rotational quantities naturally use radians, making equations more elegant and calculations more straightforward.
Gradians (also called gons or grades) divide a circle into 400 units, making right angles exactly 100 gradians. This metric-friendly system is used primarily in surveying, civil engineering, and some European countries. The advantage is easier mental math: a 1% grade equals 1 gradian, and decimal calculations are simpler than with degrees.
In standard mathematical convention, angles are measured counterclockwise from the positive x-axis. Navigation uses clockwise from north (0° = North, 90° = East). Geographic/surveying systems may use different references. Computer graphics often use different conventions. Learning about these systems is vital for GPS, mapping, and programming applications.
Coterminal angles differ by full rotations (360° or 2π radians) but share the same terminal position. For example, 45°, 405°, and -315° are coterminal. They're important because they have identical trigonometric values, help in solving equations with multiple solutions, and are essential for Learning about periodic functions and rotational motion in physics and engineering.

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Updated October 20, 2025
Published: July 19, 2025