Everyday Life · Unit Conversions
Angle Unit Converter
Converts angles between degrees, radians, gradians, arcminutes, arcseconds, and turns.
Calculator
Formula
All conversions pass through degrees as the base unit. theta_deg is the angle in degrees, theta_rad is the angle in radians, theta_grad is the angle in gradians (also called gon), and theta_turn is the angle expressed as a fraction of a full revolution. Arcminutes equal degrees divided by 60; arcseconds equal degrees divided by 3600.
Source: ISO 80000-3:2019 Quantities and units — Space and time.
How it works
Angles can be expressed in many systems depending on the discipline. The degree is the most familiar unit, dividing a full circle into 360 equal parts — a convention inherited from Babylonian base-60 arithmetic. The radian is the SI unit of angle, defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. Because a full circle has circumference 2πr, there are exactly 2π radians in 360 degrees. Radians are dimensionless and are the natural unit for calculus and physics, which is why nearly all trigonometric functions in programming languages expect radian input.
The gradian (also called gon or grad) divides a right angle into 100 parts, making a full circle 400 gradians. Introduced during the French Revolution alongside the metric system, gradians are still widely used in surveying and civil engineering across continental Europe. The turn (or revolution) expresses an angle as a fraction of a complete rotation, so one turn equals 360° or 2π radians — useful in rotational mechanics and angular velocity calculations. Arcminutes and arcseconds subdivide a degree by factors of 60 and 3600 respectively, providing fine resolution for celestial coordinates, GPS positioning, and optical instrument specifications. This converter uses degrees as an internal pivot: the input is first converted to degrees, then from degrees to every other unit, ensuring a single consistent conversion chain.
Engineers use radians when working with angular frequency (ω = 2πf), programmers must switch between degrees and radians when calling Math.sin() or Math.cos(), and astronomers routinely work in degrees, arcminutes, and arcseconds when specifying the right ascension and declination of celestial objects. Surveyors in countries using the gradian system can check their instrument readings against degree equivalents in seconds.
Worked example
Suppose a surveyor reads a horizontal angle of 150 gradians on a total station and needs to report it in degrees, radians, and arcminutes for an international team.
Step 1 — Convert gradians to degrees:
degrees = 150 × (9 / 10) = 135°
Step 2 — Convert degrees to radians:
radians = 135 × (π / 180) = 135 × 0.017453293 = 2.35619449 rad
(This is exactly 3π/4.)
Step 3 — Convert degrees to arcminutes:
arcminutes = 135 × 60 = 8,100 ′
Step 4 — Convert degrees to arcseconds:
arcseconds = 135 × 3600 = 486,000 ″
Step 5 — Convert degrees to turns:
turns = 135 / 360 = 0.375 rev (three-eighths of a full rotation)
All five results can be read directly from this calculator by entering 150 and selecting Gradians as the input unit.
Limitations & notes
This converter handles any real number, including negative angles and values beyond a full circle, which are mathematically valid in many contexts (e.g. cumulative rotation, phase angles). However, it does not normalise angles to a standard range such as 0–360° or −180° to 180°; if you need the principal value you must apply modular reduction manually. For angles expressed in degrees-minutes-seconds format (DMS, e.g. 47° 30′ 15″), you must first convert to decimal degrees before entering the value. Very large angles (e.g. millions of degrees for multi-revolution systems) are handled numerically but floating-point precision limits may cause small rounding errors in the arcseconds output. This tool does not handle solid angles (steradians), which are a distinct physical quantity.
Frequently asked questions
Why do programming languages like Python and JavaScript use radians for trigonometric functions?
Radians are the natural mathematical unit for angles because they make the derivatives of sin and cos exact: d/dx sin(x) = cos(x) holds only when x is in radians. Using degrees would require a constant factor of π/180 throughout calculus, making formulas messier. For this reason, Math.sin(), Math.cos(), and Math.tan() in JavaScript all expect radian input — always convert degrees to radians first using angle × (Math.PI / 180).
What is a gradian and who uses it?
A gradian (grad or gon) divides a right angle into 100 equal parts, so a full circle has 400 gradians. It was introduced in France during the Revolution as part of metrication. Today it remains in active use in surveying, geodesy, and civil engineering throughout continental Europe, particularly in countries such as France, Germany, and the Netherlands. The advantage is that a right angle is an exact 100 grad, simplifying arithmetic for surveyors calculating bearing corrections.
How many radians are in a full circle, and why is it 2π?
A full circle contains exactly 2π radians (approximately 6.28318 rad). This comes from the definition: one radian is the angle subtended by an arc whose length equals the radius. The full circumference of a circle is 2πr, so it spans 2πr / r = 2π radians. This relationship is why Euler's formula e^(iπ) + 1 = 0 and the period of sine and cosine functions are both expressed in terms of π.
What is the difference between arcminutes and minutes of time?
Arcminutes (′) are units of angular measure: 1 degree = 60 arcminutes = 3600 arcseconds. Minutes of time are temporal: 1 hour = 60 minutes of time. They are numerically different even though right ascension in astronomy is expressed in hours, minutes, and seconds of time rather than degrees — one hour of right ascension equals 15 degrees. This converter handles only angular arcminutes, not time-based minutes.
When should I use turns instead of degrees or radians?
Turns (also called revolutions or cycles) are most intuitive when describing repetitive or cumulative rotation — for example, a motor spinning at 3,000 RPM, a gear completing 12.5 rotations, or a phase expressed as a fraction of a full wave cycle. In digital signal processing, frequencies are sometimes given in cycles per sample (i.e. turns per sample), and the turn-based representation avoids the factor-of-2π that appears when mixing radians and cycles. Some modern mathematical texts advocate for the turn as a cleaner fundamental angle unit.
Last updated: 2025-01-15 · Formula verified against primary sources.