Converter

Degrees to radians Converter

11 months ago
Last Update on November 10, 2025
9 mins
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Convert degrees to radians and radians to degrees instantly with our free calculator. Features step-by-step calculations, comprehensive conversion formulas, reference charts, and trigonometric values for mathematics, physics, and engineering students.
Degrees to radians Converter

Degrees to Radians Conversion Calculator

Convert degrees to radians and radians to degrees with our advanced calculator. Features step-by-step calculations, comprehensive conversion formulas, reference tables, and practical examples for mathematics, physics, and engineering applications.

Degrees to Radians Conversion
Radians Result:
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Calculation Steps:
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Radians to Degrees Conversion
Degrees Result:
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Quick Conversion Examples

180 degrees
180° → π rad (3.1416)
90 degrees
90° → π/2 rad (1.5708)
45 degrees
45° → π/4 rad (0.7854)
π radians
3.1416 rad → 180°
π/2 radians
1.5708 rad → 90°
2π radians
6.2832 rad → 360°

Conversion Formulas

Degrees to Radians Formula

To convert an angle from degrees to radians, multiply the degree value by \(\frac{\pi}{180°}\). This conversion factor comes from the fundamental relationship that \(180° = \pi\) radians.

Primary Formula:

\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180°}\]

Decimal Approximation:

\[\text{Radians} = \text{Degrees} \times 0.0174533\]
Example Calculation:

Convert 60° to radians:

\[\text{Radians} = 60° \times \frac{\pi}{180°} = \frac{60\pi}{180} = \frac{\pi}{3} \approx 1.0472\text{ rad}\]

Radians to Degrees Formula

To convert an angle from radians to degrees, multiply the radian value by \(\frac{180°}{\pi}\). This is the inverse of the degrees-to-radians conversion.

Primary Formula:

\[\text{Degrees} = \text{Radians} \times \frac{180°}{\pi}\]

Decimal Approximation:

\[\text{Degrees} = \text{Radians} \times 57.2958°\]
Example Calculation:

Convert \(\frac{\pi}{4}\) radians to degrees:

\[\text{Degrees} = \frac{\pi}{4} \times \frac{180°}{\pi} = \frac{180°}{4} = 45°\]

Conversion Reference Tables

Standard Angles Conversion Chart

DegreesRadians (Exact)Radians (Decimal)Description
0°00.0000Zero angle
15°π/120.2618One-twenty-fourth rotation
30°π/60.5236One-twelfth rotation
45°π/40.7854One-eighth rotation
60°π/31.0472One-sixth rotation
90°π/21.5708Quarter rotation (right angle)
120°2π/32.0944One-third rotation
135°3π/42.3562Three-eighths rotation
150°5π/62.6180Five-twelfths rotation
180°π3.1416Half rotation (straight angle)
270°3π/24.7124Three-quarter rotation
360°2π6.2832Full rotation (complete circle)

Detailed Incremental Chart

DegreesRadiansDegreesRadiansDegreesRadians
5°0.087365°1.1345125°2.1817
10°0.174570°1.2217130°2.2689
15°0.261875°1.3090135°2.3562
20°0.349180°1.3963140°2.4435
25°0.436385°1.4835145°2.5307
30°0.523690°1.5708150°2.6180
35°0.610995°1.6581155°2.7053
40°0.6981100°1.7453160°2.7925
45°0.7854105°1.8326165°2.8798
50°0.8727110°1.9199170°2.9671
55°0.9599115°2.0071175°3.0543
60°1.0472120°2.0944180°3.1416

Trigonometric Key Angles

DegreesRadianssin θcos θtan θ
0°0010
30°π/61/2√3/2√3/3
45°π/4√2/2√2/21
60°π/3√3/21/2√3
90°π/210undefined
120°2π/3√3/2-1/2-√3
135°3π/4√2/2-√2/2-1
150°5π/61/2-√3/2-√3/3
180°π0-10

Understanding Degrees and Radians

Degrees and radians are two different measurement systems for angles, each with distinct advantages. Understanding when and why to use each system is essential for mathematics, physics, and engineering applications.

What Are Degrees?

A degree is a unit of angular measurement where one complete rotation equals 360 degrees. This system originated in ancient Babylon, where astronomers divided the circle into 360 parts, possibly because 360 has many divisors and approximates the number of days in a year. Degrees are intuitive and widely used in navigation, construction, geography, and everyday contexts.

What Are Radians?

A radian is the standard unit of angular measurement in mathematics and physics. One radian is defined as the angle subtended at the center of a circle by an arc whose length equals the circle's radius. Since the circumference of a circle equals \(2\pi r\), a full rotation equals \(2\pi\) radians (approximately 6.28318 radians). Radians provide a natural connection between angular measure and arc length.

Fundamental Relationships:

• \(360° = 2\pi\) radians
• \(180° = \pi\) radians
• \(1\) radian \(= \frac{180°}{\pi} \approx 57.2958°\)
• \(1° = \frac{\pi}{180}\) radians \(\approx 0.0174533\) radians

Why Convert Between Units?

Different contexts require different angular units for optimal clarity and calculation efficiency:

  • Calculus operations: Derivatives and integrals of trigonometric functions require radian measure for correct results
  • Physics formulas: Angular velocity, centripetal acceleration, and rotational dynamics use radians
  • Programming: Most programming languages expect radian inputs for trigonometric functions
  • Navigation: Compass bearings and geographic coordinates use degrees
  • Communication: Degrees are more intuitive for general audiences
  • Engineering: Different fields may prefer different units based on convention

Step-by-Step Conversion Guide

Method 1: Converting Degrees to Radians

Using the π Formula (Exact Method)

Step 1: Identify the angle in degrees (e.g., 135°)
Step 2: Multiply by the conversion factor \(\frac{\pi}{180°}\)
Step 3: Simplify the fraction: \(135° \times \frac{\pi}{180°} = \frac{135\pi}{180}\)
Step 4: Reduce to simplest form: \(\frac{135\pi}{180} = \frac{3\pi}{4}\) radians

Using Decimal Conversion (Approximate Method)

Step 1: Identify the angle in degrees (e.g., 135°)
Step 2: Multiply by 0.0174533 (the conversion factor)
Step 3: Calculate: \(135 \times 0.0174533 \approx 2.3562\) radians

Method 2: Converting Radians to Degrees

Using the π Formula (Exact Method)

Step 1: Identify the angle in radians (e.g., \(\frac{5\pi}{6}\))
Step 2: Multiply by the conversion factor \(\frac{180°}{\pi}\)
Step 3: Simplify by canceling π: \(\frac{5\pi}{6} \times \frac{180°}{\pi} = \frac{5 \times 180°}{6}\)
Step 4: Calculate: \(\frac{900°}{6} = 150°\)

Using Decimal Conversion (Approximate Method)

Step 1: Identify the angle in radians (e.g., 2.618)
Step 2: Multiply by 57.2958 (the conversion factor)
Step 3: Calculate: \(2.618 \times 57.2958 \approx 150°\)

Practical Applications

Mathematics and Trigonometry

Angle conversion is fundamental to trigonometric calculations and mathematical analysis:

  • Unit circle: Understanding relationships between degrees and radians helps visualize trigonometric functions
  • Function evaluation: Calculators require correct mode (DEG or RAD) for accurate results
  • Periodic functions: Period calculations often involve converting between units
  • Angle identities: Many trigonometric identities work more elegantly in radians

Calculus and Analysis

Calculus operations mandate radian measure for standard formulas to work correctly:

  • Derivatives: \(\frac{d}{dx}\sin(x) = \cos(x)\) only holds when \(x\) is in radians
  • Integrals: \(\int \cos(x)dx = \sin(x) + C\) requires radian measure
  • Series expansions: Taylor series for trigonometric functions use radians
  • Limits: The fundamental limit \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\) is true only for radians

Physics and Engineering

Physical formulas involving rotation and oscillation inherently use radians:

  • Angular velocity: \(\omega = \frac{\Delta\theta}{\Delta t}\) measured in rad/s
  • Arc length: \(s = r\theta\) works directly with radians (no conversion factor needed)
  • Rotational kinetic energy: \(KE = \frac{1}{2}I\omega^2\) uses angular velocity in rad/s
  • Simple harmonic motion: \(x(t) = A\cos(\omega t + \phi)\) with \(\omega\) in rad/s
  • Centripetal acceleration: \(a_c = r\omega^2\) requires radians for \(\omega\)

Computer Programming

Programming applications frequently require angle conversion:

  • Graphics rendering: 3D rotations and transformations typically use radians internally
  • Game development: Character rotation, camera angles, and physics calculations
  • Animation: Smooth interpolation between angles requires consistent units
  • Trigonometric functions: Math libraries (Math.sin, Math.cos) expect radian inputs
  • User interfaces: Displaying angles to users often requires degrees for readability

Navigation and Surveying

Geographic and directional applications predominantly use degrees:

  • Compass bearings: Traditional navigation uses degrees (0° - 360°)
  • Latitude and longitude: Geographic coordinates measured in degrees
  • Survey measurements: Land surveying typically records angles in degrees
  • Aviation: Flight headings specified in degrees
  • GPS systems: Display coordinates in degrees but may calculate internally with radians

Common Conversion Scenarios

Converting Standard Angles

ScenarioDegreesRadians (Exact)Radians (Decimal)
Acute angle (small)30°π/60.5236
Acute angle (medium)45°π/40.7854
Acute angle (large)75°5π/121.3090
Right angle90°π/21.5708
Obtuse angle135°3π/42.3562
Straight angle180°π3.1416
Reflex angle270°3π/24.7124
Full rotation360°2π6.2832

Tips and Best Practices

For Students

  • Memorize key conversions: Know common angle equivalents (30°, 45°, 60°, 90°, 180°)
  • Learn exact forms: Recognize fractions of π (π/6, π/4, π/3, π/2) and their degree equivalents
  • Check calculator mode: Verify DEG or RAD mode before solving problems
  • Leave answers in terms of π: For exact work, express radians as multiples of π
  • Simplify fractions: Always reduce fractional coefficients of π to simplest form
  • Unit awareness: Pay attention to which unit problems expect in final answers

For Programmers

  • Use constants: Leverage Math.PI or equivalent constants for accuracy
  • Create helper functions: Write reusable deg2rad and rad2deg conversion functions
  • Default to radians: Standard math libraries expect radian inputs
  • Comment units clearly: Document whether angles are in degrees or radians
  • Consider user input: Users typically think in degrees; convert internally to radians
  • Handle edge cases: Account for angles outside standard ranges [0, 2π) or [0°, 360°)

For Engineers

  • Maintain consistency: Use one unit system throughout a project when possible
  • Document assumptions: Clearly state angular units in technical specifications
  • Verify software settings: Check default units in CAD, simulation, and analysis tools
  • SI units preference: Radians are the SI unit for angular measurement
  • Communication clarity: Use degrees when presenting to non-technical audiences
Common Mistakes to Avoid:

• Forgetting to convert before using trigonometric functions
• Using degrees in calculus formulas that require radians
• Calculator in wrong mode (DEG vs RAD)
• Rounding π too early in calculations (use at least 3.14159)
• Not simplifying fractional multiples of π
• Mixing degrees and radians in the same calculation
• Forgetting the conversion factor (multiply vs. divide)
• Not checking units in physics formulas

Frequently Asked Questions

Why do we use radians in calculus instead of degrees?
Radians make calculus formulas work naturally without correction factors. The derivative of sin(x) equals cos(x) only when x is in radians. With degrees, an extra factor of π/180 would appear in every derivative and integral of trigonometric functions, making calculations unnecessarily complex. Radians also connect directly to arc length through the formula s = rθ, creating mathematical elegance that degrees cannot match.
How do I know if my calculator is in degree or radian mode?
Most scientific calculators display "DEG" or "RAD" on the screen to indicate the current mode. Check your calculator's manual for the specific button to toggle between modes (often labeled "MODE" or "DRG"). To test, calculate sin(90): if the result is 1, you're in degree mode; if it's approximately 0.894, you're in radian mode. Always verify the mode before starting calculations to avoid errors.
Can I convert negative angles or angles greater than 360°?
Yes, the conversion formulas work for all angle values. Negative angles represent clockwise rotation, while angles greater than 360° (or 2π radians) indicate multiple complete rotations. For example, -45° converts to -π/4 radians, and 450° converts to 5π/2 radians. In many applications, angles are normalized to the range [0°, 360°) or [0, 2π) by adding or subtracting full rotations, but the conversion process remains the same.
Should I memorize the conversion factors?
Yes, memorizing the key relationships saves time and reduces errors. Remember that π radians = 180°, which gives the conversion factors: multiply by π/180 to convert degrees to radians, and multiply by 180/π to convert radians to degrees. Also memorize decimal approximations: multiply by 0.0174533 (deg to rad) or by 57.2958 (rad to deg). Most importantly, remember common angle equivalents like π/2 = 90° and π/4 = 45°.
Why does 2π equal 360° and not a simpler number?
The relationship between π and 360° comes from two independent historical choices. The number 360 for a full circle was chosen by ancient Babylonians for its many divisors and approximate match to the solar year. The constant π (approximately 3.14159) is the mathematical ratio of circumference to diameter, independent of the degree system. Since the circumference is 2πr and one radian equals the angle for arc length r, a full circle contains 2π radians. These two systems developed separately and happen to relate through this π-based conversion.
Do programming languages use degrees or radians?
Most programming languages (Python, JavaScript, C++, Java, etc.) default to radians for trigonometric functions. Math.sin(x), Math.cos(x), and similar functions expect x to be in radians. Some languages provide conversion functions or degree-based alternatives (like Python's math.degrees() and math.radians()), but the standard trigonometric functions universally use radians. Always convert degree inputs to radians before passing them to these functions, or use degree-specific functions if available.
What's the most accurate way to convert between degrees and radians?
For exact mathematical work, use the π-based formulas and leave answers in terms of π (e.g., π/4 rather than 0.7854). When numerical values are required, use a high-precision value of π (at least 3.14159, ideally more). Programming languages provide built-in π constants with machine precision (typically 15-17 significant digits). Avoid chaining multiple conversions, as each step introduces potential rounding error. For critical applications, consider using symbolic math systems that handle π exactly.
Why are radians considered "natural" units?
Radians are natural because they arise directly from circle geometry without arbitrary scaling. One radian is the angle that creates an arc equal to the radius—a pure geometric relationship. This makes radians dimensionless (a ratio of two lengths) and allows formulas like s = rθ to work without conversion factors. Degrees, by contrast, arbitrarily divide circles into 360 parts. The naturalness of radians becomes especially clear in calculus, where derivatives and series expansions work seamlessly with radians but require correction factors with degrees.
How do I convert degrees-minutes-seconds (DMS) to radians?
First convert DMS format to decimal degrees: decimal degrees = degrees + (minutes/60) + (seconds/3600). For example, 45°30'15" = 45 + (30/60) + (15/3600) = 45.5042°. Then multiply by π/180 to convert to radians: 45.5042° × (π/180) ≈ 0.7941 radians. Many calculators and online tools can perform this two-step conversion automatically. Remember that 1 minute = 1/60 degree and 1 second = 1/3600 degree.
Is there ever a situation where degrees are better than radians?
Yes, degrees are often preferable for practical communication and certain applications. Navigation and aviation use degrees for compass headings because 0°-360° is intuitive. Geographic coordinates (latitude/longitude) use degrees for historical reasons and human readability. Construction and surveying use degrees because they're easier to visualize and communicate. For teaching introductory trigonometry, degrees help students build intuition before introducing radians. However, for any calculation involving calculus, physics formulas, or advanced mathematics, radians are essential.

Historical Development

The Degree System

The 360-degree circle division originated with ancient Babylonian astronomers around 3000 BCE. Working with a base-60 (sexagesimal) number system, they chose 360 because it has 24 divisors, making it easy to divide circles into halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths, twelfths, and more without fractions. Additionally, 360 approximates the number of days in a year, connecting angular measurement to celestial cycles.

The Development of Radians

Radians emerged much later as mathematicians sought more natural units for mathematical analysis. Roger Cotes (1682-1716) and Leonhard Euler (1707-1783) used radian-based concepts in their work, though the term "radian" wasn't formally adopted until 1873 by James Thomson. The unit gained acceptance because it simplified calculus formulas and provided a direct connection between angular measure and arc length.

Modern Standardization

Today, the International System of Units (SI) recognizes the radian as the standard unit for angular measurement in scientific and engineering contexts. Degrees remain acceptable for practical applications and communication, but radians dominate in theoretical work, physics, and higher mathematics.

Mathematical Foundations

Arc Length Relationship

The fundamental definition of a radian connects angle measurement directly to arc length. For a circle with radius \(r\), an angle \(\theta\) (in radians) subtends an arc of length:

\[s = r\theta\]

This elegant formula works only when \(\theta\) is measured in radians. With degrees, the formula requires a conversion factor: \(s = r\theta \times \frac{\pi}{180}\), making calculations more complex.

Calculus Prerequisites

The fundamental limits and derivatives that underpin trigonometric calculus require radian measure:

\[\lim_{\theta \to 0} \frac{\sin(\theta)}{\theta} = 1\] (only true for radians)

\[\frac{d}{d\theta}\sin(\theta) = \cos(\theta)\] (only true for radians)

\[\frac{d}{d\theta}\cos(\theta) = -\sin(\theta)\] (only true for radians)

With degrees, these relationships require multiplication by \(\frac{\pi}{180}\), destroying the mathematical elegance and complicating all subsequent work.

Dimensionless Property

Unlike degrees, radians are dimensionless—they represent a pure ratio of two lengths (arc length to radius). This property makes radians mathematically cleaner and eliminates confusion about units in complex formulas. The dimensionless nature explains why radians often appear in equations without explicit unit labels.

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