Convert between different angle units including degrees, radians, gradians, and turns. Essential for mathematics, physics, engineering, and trigonometry. Get instant conversions between any angle measurement units.
Understanding Angle Measurement
Angles measure the rotation between two lines that share a common endpoint (vertex). They appear throughout mathematics, engineering, navigation, astronomy, and everyday life. While most people are familiar with degrees from school geometry, several other angle units are used in specific professional contexts, and converting between them accurately is important for technical work.
Degrees: The Everyday Standard
The degree system divides a full rotation into 360 equal parts. This number was chosen by ancient Babylonian astronomers, likely because 360 is close to the number of days in a year and is highly divisible (by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180). Key reference angles: 90° (right angle), 180° (straight line), 360° (full rotation). Degrees are further subdivided into arcminutes (1/60 of a degree) and arcseconds (1/60 of an arcminute), used in precise navigation and astronomy.
Radians: The Mathematical Standard
Radians are the standard unit of angle measurement in mathematics and physics. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius. A full circle contains exactly 2π radians (approximately 6.2832 radians). The conversion: 1 radian = 180/π degrees ≈ 57.296°. Radians simplify mathematical formulas — particularly in calculus, wave functions, and trigonometry. When no unit is specified in a mathematical formula involving trigonometric functions, radians are assumed.
Common Angle Conversions
- 90° = π/2 radians (right angle)
- 180° = π radians (straight line)
- 360° = 2π radians (full rotation)
- 1° = π/180 ≈ 0.01745 radians
- 1 radian ≈ 57.296°
- Gradians: 400 gradians = 360° (used in surveying)
Gradians and Turns
Gradians (also called grads or gon) divide a full circle into 400 equal parts, making a right angle exactly 100 gradians — convenient for surveying and civil engineering calculations. Some scientific calculators include a grad mode alongside degrees and radians. A turn (or revolution) represents a complete 360° rotation and is sometimes used in engineering to describe shaft rotations or winding counts.
Frequently Asked Questions
When should I use radians instead of degrees?
Use radians when working with mathematical formulas, calculus, physics equations, or programming trigonometric functions. Most programming languages' math libraries (JavaScript Math.sin(), Python math.sin(), etc.) accept angles in radians by default — passing degrees gives wrong results. Use degrees for everyday geometry, navigation, architecture, and any context where people expect familiar 0-360 notation.
What is the difference between an angle and its supplement or complement?
Complementary angles sum to 90° (π/2 radians) — for example, 30° and 60° are complementary. Supplementary angles sum to 180° (π radians) — for example, 45° and 135° are supplementary. Explementary (or conjugate) angles sum to 360° (2π radians). These relationships appear throughout geometry, trigonometry, and physics, particularly in optics, mechanics, and wave analysis.