Angle Conversion

Convert between degrees, radians and gradians for mathematical calculations

Angle Conversion Formulas

Degrees ↔ Radians

Radians = Degrees × (π/180)

Degrees = Radians × (180/π)

π ≈ 3.14159265359

Degrees ↔ Gradians

Gradians = Degrees × (10/9)

Degrees = Gradians × (9/10)

400 gradians = 360 degrees

Radians ↔ Gradians

Gradians = Radians × (200/π)

Radians = Gradians × (π/200)

2π radians = 400 gradians

Key Conversion Values

180° = π radians = 200 gradians

90° = π/2 radians = 100 gradians

45° = π/4 radians = 50 gradians

30° = π/6 radians ≈ 33.33 gradians

Common angles in all systems

Full Circle Relationships

360° = 2π radians = 400 gradians

1° ≈ 0.01745 radians ≈ 1.111 gradians (or π/180 radians or 10/9 gradians)

1 radian ≈ 57.2958° ≈ 63.662 gradians (or 180/π degrees or 200/π gradians)

1 gradian = 0.9° ≈ 0.01571 radians (or 9/10 degrees or π/200 radians)

Complete rotation equivalences

Reference Angle Values

Common Angle Degrees (°) Radians (rad) Decimal Radians Gradians (ᵍ) sin θ cos θ
Zero 0° 0 0.0000 0ᵍ 0 1
Thirty degrees 30° π/6 0.5236 33.33ᵍ 1/2 √3/2
Forty-five degrees 45° π/4 0.7854 50ᵍ √2/2 √2/2
Sixty degrees 60° π/3 1.0472 66.67ᵍ √3/2 1/2
Right angle 90° π/2 1.5708 100ᵍ 1 0
Straight angle 180° π 3.1416 200ᵍ 0 -1
Three-quarter turn 270° 3π/2 4.7124 300ᵍ -1 0
Full circle 360° 2π 6.2832 400ᵍ 0 1

Angle Unit Definitions

Radian (rad) - Mathematical Standard

The natural unit of angular measurement, defined as the angle subtended by an arc equal in length to the radius.

Definition: Angle where arc length equals radius length

Mathematical Basis: Derived from the geometry of circles

SI Status: SI derived unit, dimensionless

Usage: Calculus, physics, engineering calculations

Advantage: Simplifies many mathematical formulas

Degree (°) - Traditional Standard

Historical unit dividing a full circle into 360 equal parts.

Definition: 1/360 of a complete rotation

Historical Origin: Babylonian base-60 system and approximate days in a year

Usage: Navigation, surveying, everyday measurements

Subdivisions: Minutes (') and seconds (") for precision

Intuitive: Familiar and easy to visualize

Gradian (ᵍ or gon) - Decimal Standard

A metric unit dividing a full circle into 400 equal parts, designed for decimal convenience.

Definition: 1/400 of a complete rotation

Historical Origin: Introduced during the French Revolution as part of metric system

Usage: Surveying, civil engineering, some European countries

Advantage: Right angle = 100 gradians (decimal-friendly)

Also Known As: Gon, grade, or grad

Minutes and Seconds

Arc Minute ('): 1/60 of a degree = 1/21,600 of a full circle

Arc Second ("): 1/60 of a minute = 1/1,296,000 of a full circle

Usage: Astronomy, navigation, precise angle measurements

Example: 45° 30' 15" = 45.5042 degrees

Other Angle Units

Turns (rev): Complete rotations, used in mechanical engineering

Mils: Military unit, 1/6400 of a full circle

Compass Bearings: 0-360° from north for navigation

Hour Angle: 1/24 of a circle, used in astronomy

Scientific Applications

Physics & Engineering

Rotational Mechanics: Angular momentum L = Iω

Wave Physics: Phase angles, frequency analysis

Control Systems: Phase margins, stability analysis

Mathematics & Calculus

Derivatives: d/dx(sin x) = cos x (x in radians)

Series Expansions: sin x = x - x³/6 + x⁵/120 - ...

Complex Analysis: e^(iθ) = cos θ + i sin θ

Computer Graphics

Rotation Matrices: 2D and 3D transformations

Animation: Smooth rotational interpolation

Game Development: Object orientation and movement

Astronomy & Navigation

Celestial Coordinates: Right ascension, declination

GPS Navigation: Bearing calculations

Stellar Positioning: Parallax measurements

Mechanical Engineering

Gear Design: Tooth angles and meshing

Robotics: Joint angles and kinematics

Machine Tools: Cutting angles and tool positioning

Signal Processing

Fourier Analysis: Frequency domain representation

Phase Relationships: Signal timing and synchronization

Filter Design: Phase response characteristics

Calculation Considerations

Unit System Selection

Mathematical Work: Use radians for calculus and analytical work

Practical Applications: Degrees for engineering drawings and navigation

Programming: Most math libraries expect radians as input

Precision Considerations

Floating Point: Be aware of rounding errors in conversions

Very Small Angles: sin θ ≈ θ for small θ in radians

Periodic Functions: Remember trigonometric function periodicity

Common Errors

Calculator Mode: Ensure calculator is in correct angle mode

Programming: Check if functions expect degrees or radians

Mixed Units: Avoid mixing degrees and radians in calculations

Fundamental Angle Equations

Arc Length Relationship

s = rθ

Where: s = arc length (m), r = radius (m), θ = angle (radians). Only valid when θ is in radians.

Angular Velocity

ω = θ/t = 2πf

Where: ω = angular velocity (rad/s), θ = angle (radians), t = time (s), f = frequency (Hz)

Trigonometric Functions

sin²θ + cos²θ = 1

tan θ = sin θ / cos θ

Fundamental trigonometric identities

Angular Acceleration

α = dω/dt = d²θ/dt²

Where: α = angular acceleration (rad/s²), ω = angular velocity (rad/s), θ = angle (radians)