Angle Calculator

Convert angles between degrees, radians, and gradians

Enter an angle value and select the conversion type to instantly convert between different angle units. Supports all common angle measurements.

Examples

  • 90° = π/2 ≈ 1.571 radians = 100 gradians
  • π radians = 180° = 200 gradians
  • 45° = π/4 ≈ 0.785 radians = 50 gradians
  • 1 radian ≈ 57.296° ≈ 63.662 gradians

Important Note

Degrees are most common (360° in circle), radians are used in calculus (2π in circle), and gradians divide the circle into 400 units.

Other Titles
Understanding Angle Calculator: A Comprehensive Guide
Explore angle measurement units, their conversions, applications in mathematics, engineering, and navigation systems

Understanding Angle Calculator: A Comprehensive Guide

  • Different angle units serve various mathematical and practical purposes
  • Conversions between units are essential in trigonometry and calculus
  • Understanding angle measurements improves mathematical problem-solving
Angles can be measured in several different units, each with specific applications and advantages. The three most common units are degrees, radians, and gradians.
Degrees divide a full circle into 360 equal parts and are most familiar in everyday use. Radians are based on the circle's radius and are essential in calculus and advanced mathematics. Gradians divide a circle into 400 parts and are used in some surveying applications.
Converting between these units is crucial for solving problems across different mathematical contexts and ensuring compatibility between various systems and applications.
Our angle calculator provides instant, accurate conversions between all three major angle measurement systems.

Basic Angle Conversions

  • Right angle: 90° = π/2 rad ≈ 1.571 rad = 100 grad
  • Straight angle: 180° = π rad ≈ 3.142 rad = 200 grad
  • Full circle: 360° = 2π rad ≈ 6.283 rad = 400 grad

Step-by-Step Guide to Using the Angle Calculator

  • Learn how to select the correct conversion type
  • Understand input requirements for each angle unit
  • Master interpreting and applying conversion results
Our angle calculator simplifies the conversion process between degrees, radians, and gradians with just a few simple steps.
Step 1: Choose Conversion Type
Select the appropriate conversion from the dropdown menu based on your input and desired output units.
Step 2: Enter the Angle Value
Input your angle measurement. The calculator accepts positive and negative values, decimals, and large numbers.
Step 3: Review the Result
The converted angle appears instantly with high precision. Use this result in your calculations or copy it for use in other applications.
Common Conversion Formulas:
  • Degrees to Radians: multiply by π/180
  • Radians to Degrees: multiply by 180/π
  • Degrees to Gradians: multiply by 10/9

Calculator Usage Examples

  • Convert 60°: Select 'Degrees to Radians' → Result: π/3 ≈ 1.047 radians
  • Convert 2π rad: Select 'Radians to Degrees' → Result: 360°
  • Convert 150 grad: Select 'Gradians to Degrees' → Result: 135°

Real-World Applications of Angle Calculator Conversions

  • Engineering: CAD systems and technical drawings
  • Navigation: GPS systems and aviation
  • Mathematics: Calculus and trigonometric functions
  • Science: Physics calculations and astronomical measurements
Angle conversions are essential across numerous fields and applications:
Engineering and Design:
  • CAD Software: Different programs may use different angle units, requiring conversions for compatibility.
  • Mechanical Engineering: Gear ratios and rotational mechanics often involve radian measurements.
Navigation and Surveying:
  • GPS Systems: Coordinate systems may use different angular units for latitude and longitude calculations.
  • Land Surveying: Gradians are sometimes used in European surveying practices.
Mathematics and Science:
  • Calculus: Derivatives and integrals of trigonometric functions require radian measure.
  • Physics: Angular velocity and acceleration calculations typically use radians per second.

Practical Applications

  • Robot arm rotation: Convert 45° to 0.785 radians for motor control
  • Satellite dish alignment: Convert bearing from degrees to gradians for European equipment
  • Pendulum physics: Convert angle from degrees to radians for period calculations
  • Astronomy: Convert celestial coordinates between different angular systems

Common Misconceptions and Correct Methods in Angle Conversions

  • Addressing frequent errors in unit identification
  • Clarifying when to use each angle measurement system
  • Explaining precision requirements in different applications
Understanding common misconceptions about angle units helps ensure accurate conversions and appropriate unit selection.
Misconception 1: All Calculators Use the Same Angle Mode
Wrong: Assuming calculators automatically know which angle unit you're using.
Correct: Always check your calculator's angle mode (DEG/RAD/GRAD) before performing trigonometric calculations.
Misconception 2: Radians Are Only for Advanced Mathematics
Wrong: Thinking radians are unnecessary for practical applications.
Correct: Many engineering and physics applications naturally use radians, making them essential for technical work.
Misconception 3: Approximate Conversions Are Sufficient
Wrong: Using rough approximations like π ≈ 3.14 for precision calculations.
Correct: Use full precision (π) in calculations and round only final results when appropriate.

Common Error Corrections

  • Wrong: Using sin(30) in degree mode expecting sin(30°) result
  • Correct: Set calculator to degree mode first, then calculate sin(30°)
  • Wrong: Converting 1 radian ≈ 60 degrees
  • Correct: 1 radian = 180/π ≈ 57.296 degrees

Mathematical Derivation and Examples

  • Understanding the mathematical basis of angle unit definitions
  • Deriving conversion formulas from fundamental principles
  • Connecting angle measures to circle geometry
The relationships between angle units stem from different ways of dividing a circle and mathematical conventions.
Fundamental Definitions:
  • Degree: 1/360 of a full circle (360° = full circle)
  • Radian: Angle subtended by arc length equal to radius (2π rad = full circle)
  • Gradian: 1/400 of a full circle (400 grad = full circle)
Conversion Formula Derivation:
Since 360° = 2π rad = 400 grad for a full circle:
  • Degrees to Radians: θ(rad) = θ(deg) × π/180
  • Radians to Degrees: θ(deg) = θ(rad) × 180/π
  • Degrees to Gradians: θ(grad) = θ(deg) × 400/360 = θ(deg) × 10/9

Mathematical Foundations

  • Verification: 180° = π rad because 180 × π/180 = π ✓
  • Circle arc: Arc length = radius × angle(radians) shows radian definition
  • Unit circle: Angle in radians equals arc length on unit circle
  • Surveying: 100 grad = 90° shows right angle in gradian system