Angle Converter
Convert between degrees, radians, gradians, and other angle units
All Converters
Common Angle Conversions
Angles in Real Life
Clock Hands
Each hour mark = 30° (π/6 rad)
Right Angle
Corner of square = 90° (π/2 rad)
Compass Bearing
North = 0°, East = 90°, South = 180°
Triangle Sum
All angles sum to 180° (π rad)
About Angle Measurement
Angle measurement is a fundamental concept in mathematics, physics, engineering, and navigation that quantifies the amount of rotation or the space between two intersecting lines. Understanding angles is essential for everything from basic geometry to advanced applications in astronomy, robotics, and computer graphics.
What is an Angle?
An angle is formed when two rays (or lines) meet at a common point called the vertex. The magnitude of an angle is measured by the amount of rotation needed to bring one ray into coincidence with the other. In mathematical terms, an angle θ is defined as the ratio of the arc length s to the radius r of a circle:
θ = s/r (in radians)
This definition forms the basis for all angle measurements and conversions between different units.
Common Angle Units and Conversions
| Unit | Symbol | Definition | Conversion to Degrees |
|---|---|---|---|
| Degree | ° | 1/360 of a full circle | 1° |
| Radian | rad | Central angle subtending arc equal to radius | 180°/π ≈ 57.2958° |
| Gradian | grad | 1/400 of a full circle | 0.9° |
| Arc Minute | ′ | 1/60 of a degree | 1/60° |
| Arc Second | ″ | 1/60 of an arc minute | 1/3600° |
| Turn | turn | Complete rotation (full circle) | 360° |
Angle Types and Classifications
| Angle Type | Range | Real-World Examples |
|---|---|---|
| Acute | 0° < θ < 90° | Clock hands at 2:00, scissors blades, mountain slopes |
| Right | θ = 90° | Corner of square, perpendicular walls, clock at 3:00 |
| Obtuse | 90° < θ < 180° | Clock hands at 4:00, open book, wide V-shape |
| Straight | θ = 180° | Straight line, clock at 6:00, flat surface |
| Reflex | 180° < θ < 360° | Clock hands at 8:00, wide turn, large arc |
| Full Turn | θ = 360° | Complete circle, full rotation, clock at 12:00 |
Degrees vs Radians: Mathematical Relationships
The relationship between degrees and radians is fundamental to trigonometry and calculus. The conversion formulas are:
Degrees to Radians
θ(rad) = θ(°) × (π/180)
Example: 90° = 90 × (π/180) = π/2 rad
Radians to Degrees
θ(°) = θ(rad) × (180/π)
Example: π/4 rad = π/4 × (180/π) = 45°
Frequently Asked Questions About Angle Conversion
Related Angle Conversion Tools
Length Converter
Convert between different length units including arc length calculations
Area Converter
Calculate areas including circular sectors and angular measurements
Angular Velocity
Convert angular velocity units like radians per second and degrees per minute
Angular Acceleration
Convert angular acceleration units for rotational motion calculations
Moment of Inertia
Calculate rotational inertia which depends on angular measurements
Torque Converter
Convert torque units which involve angular force measurements