Angle Converter
Convert between degrees, radians, and grads with step-by-step calculations.
Scroll down for angle reference charts
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Understanding Angle Conversions
A Comprehensive Guide to Angle Measurements and Conversions
Angle conversion is fundamental in mathematics, physics, engineering, navigation, and computer graphics. Whether you're working with geometric calculations, trigonometry, or rotational mechanics, understanding how to convert between degrees, radians, and grads is essential. This guide will help you master angle conversions across all applications.
Degrees (°)
Convert using the most common angle unit. 360° in a full circle, perfect for navigation and everyday measurements.
Radians (rad)
Work with the mathematical standard. 2π radians in a full circle, essential for calculus and advanced mathematics.
Grads (gon)
Convert using the decimal system. 400 grads in a full circle, used in surveying and some European countries.
Cross-Unit Conversion
Convert between different angle systems. Essential for international projects and scientific applications.
Precision Angles
Work with precise angular measurements for engineering, astronomy, and scientific applications.
Large Angle Handling
Convert angles greater than 360° and understand principal angles for rotational mechanics and physics.
Master Angle Conversions with Real Examples
Learn step-by-step how to convert between different angle units with practical examples from trigonometry, navigation, and engineering
1. Degrees (°) Conversions
Degree Conversion = Value × Conversion Factor
Example: 90° to radians → 90 × (π ÷ 180) = π/2 rad
Key Factors: To radians (×π/180), To grads (×400/360)
360° = 2π radians = 400 grads (full circle)
Purpose:
Converts from degrees to other angle units.
“How many radians are in 45 degrees?”
Benefits:
Most intuitive unit for everyday use. Common in navigation, construction, and basic geometry.
Steps:
1. Identify target unit (radians or grads)
2. Apply conversion factor (π/180 for radians, 400/360 for grads)
3. Calculate and add the correct unit
Example: 45° to radians
Calculation: 45 × (π ÷ 180) = π/4
Result: π/4 ≈ 0.785 radians
Result:
45 degrees = π/4 radians ≈ 0.785 radians
Example:
“A 45° angle in a right triangle equals π/4 radians, which is exactly half a right angle.”
2. Radians (rad) Conversions
Radian Conversion = Value × Conversion Factor
Example: π/2 rad to degrees → (π/2) × (180 ÷ π) = 90°
Key Factors: To degrees (×180/π), To grads (×400/2π)
2π radians = 360° = 400 grads (full circle)
Purpose:
Converts from radians to other angle units.
“How many degrees are in π/3 radians?”
Benefits:
Natural unit for calculus and trigonometry. Simplifies mathematical formulas and derivatives.
Steps:
1. Identify target unit (degrees or grads)
2. Apply conversion factor (180/π for degrees, 400/2π for grads)
3. Calculate and add the correct unit
Example: π/3 radians to degrees
Calculation: (π/3) × (180 ÷ π) = 60
Result: 60 degrees
Result:
π/3 radians = 60 degrees
Example:
“π/3 radians is the angle in an equilateral triangle, which equals exactly 60 degrees.”
3. Grads (gon) Conversions
Grad Conversion = Value × Conversion Factor
Example: 100 gon to degrees → 100 × (360 ÷ 400) = 90°
Key Factors: To degrees (×360/400), To radians (×2π/400)
400 grads = 360° = 2π radians (full circle)
Purpose:
Converts from grads to other angle units.
“How many degrees are in 200 grads?”
Benefits:
Decimal-based system makes calculations easier. Used in surveying and some European countries.
Steps:
1. Identify target unit (degrees or radians)
2. Apply conversion factor (360/400 for degrees, 2π/400 for radians)
3. Calculate and add the correct unit
Example: 200 grads to degrees
Calculation: 200 × (360 ÷ 400) = 180
Result: 180 degrees
Result:
200 grads = 180 degrees
Example:
“200 grads is exactly half a circle, equivalent to a straight line or 180 degrees.”
4. Large Angle Conversions
Large Angle = Multiple Rotations + Principal Angle
Example: 450° = 1 full rotation + 90° (principal angle)
Principal Angle: angle mod 360° (or 2π rad, or 400 gon)
The equivalent angle within one full rotation
Purpose:
Handles angles greater than one full rotation.
“What is the principal angle of 720°?”
Benefits:
Essential for rotational mechanics, periodic functions, and understanding angular motion in physics and engineering.
Steps:
1. Convert the angle normally using standard factors
2. Identify how many full rotations are included
3. Find the principal angle (remainder after full rotations)
Example: 450° to radians
Calculation: 450 × (π ÷ 180) = 2.5π rad
Result: 2.5π rad (1.25 full rotations)
Result:
450° = 2.5π radians (Principal: π/2 rad or 90°)
Example:
“A wheel that rotates 450° has made 1.25 full turns, ending at the same position as a 90° rotation.”
Tips & Best Practices for Angle Conversions
Remember: π ≈ 3.14159, so π/180 ≈ 0.01745 (deg to rad)
Memorize common conversions: 90° = π/2 rad, 180° = π rad, 360° = 2π rad
Use exact values when possible: π/4, π/3, π/2 instead of decimals
Exact values preserve mathematical relationships and avoid rounding errors.
Essential Tips:
- • Memorize key angles: 30°, 45°, 60°, 90°, 180°, 360°
- • Know π ≈ 3.14159 for quick mental calculations
- • Use dimensional analysis to check your work
- • Remember: smaller numbers for radians, larger for degrees
- • Practice with common trigonometric angles
Best Practices:
- • Use radians for calculus and advanced mathematics
- • Use degrees for navigation and everyday measurements
- • Consider grads for surveying and decimal-based work
- • Always specify units in your final answer
- • Use conversion tools for complex calculations
Real-Life Applications:
- • Navigation and GPS systems
- • Computer graphics and game development
- • Engineering and mechanical design
- • Astronomy and satellite tracking
- • Physics and rotational mechanics
Watch Out For:
- • Confusing degrees and radians in calculations
- • Using calculator in wrong angle mode
- • Forgetting π in radian conversions
- • Not considering principal angles for large values
- • Mixing up conversion factors (×π/180 vs ×180/π)