Angle Converter (General)

The calculator for angle unit conversion converts between all major angular measurement units — degrees, radians, gradians, arcminutes, arcseconds, milliradians, and revolutions — instantly and in any direction. Different disciplines use different angular units, and this converter eliminates the arithmetic errors that arise when translating between them.

The Six Major Angle Units

Angular measurement uses six main unit systems with different bases:

• Degrees (°): 360° in a full circle; the most familiar unit; used in navigation, geography, and everyday applications
• Radians (rad): 2π radians in a full circle; the natural unit for mathematics and physics; arc length = radius × angle in radians
• Gradians (gon): 400 gradians in a full circle; used in surveying and civil engineering; 100 gradians = right angle
• Arcminutes (′): 60 arcminutes = 1 degree; used in navigation, astronomy, and precision angle measurement
• Arcseconds (″): 60 arcseconds = 1 arcminute; used in astronomy (stellar parallax), geodesy, and precision optics
• Revolutions (rev): 1 revolution = 360° = 2π radians; used for rotational speed and cycle counting

Use this online calculator for any angle unit conversion. The degrees to radians calculator and radians to degrees calculator provide focused converters for the most common conversion pair.

Radians: Why Mathematics Uses Them

Radians are the preferred unit in mathematics and physics because they make trigonometric derivatives and series expansions simple. The derivative of sin(x) is cos(x) only when x is in radians — in degrees, an extra conversion factor of π/180 appears throughout every derivative. All Taylor series for trig functions (sin x = x − x³/6 + ...) are valid only in radians. Arc length s = rθ and sector area A = ½r²θ are both simple when θ is in radians. This simplicity is why radians are the SI unit for plane angle despite being less intuitive than degrees for everyday use.

Arcminutes and Arcseconds in Astronomy and Navigation

Sub-degree angle units are critical in astronomy and navigation. One arcsecond corresponds to approximately 4.85 × 10⁻⁶ radians — a tiny angle, yet measurable with modern instruments. Stellar parallax (measuring star distances by Earth's orbital shift) was first measured in arcseconds in 1838. The parsec, the standard unit of astronomical distance, is defined as the distance at which 1 AU subtends exactly 1 arcsecond of parallax. In navigation, 1 arcminute of latitude corresponds to exactly 1 nautical mile — the practical basis of marine navigation. The angle converters category provides the complete set of angle unit conversion tools.

Gradians in Surveying and Civil Engineering

The gradian (also called gon or grad) divides the right angle into 100 equal parts, making horizontal and vertical angles multiples of simple numbers in surveying calculations. A right angle = 100 gradians; a straight line = 200 gradians; a full circle = 400 gradians. This decimal structure simplifies calculations in land surveying, especially when computing slope percentages from angular measurements. European surveying instruments often display angles in gradians by default, which requires conversion when working with international data sets.