Angle Converter — Degrees, Radians, Gradians, Arcseconds, Mils: Free Angle Converter tool: convert angles across all coordinate systems. Instant conversion between Degrees (°), Radians (rad), Gradians (grad).
Primary Use
Unit conversion across metric, imperial, and scientific standards
Input Formats
Quantities with source measurement units
Output Format
Target converted units with exact conversion factors
Processing Mode
100% Client-Side In-Browser Conversion
Access Tier
Free, unlimited, no sign-up needed
The Mathematics of Angular Measurement & Coordinate Systems
An angle is the geometric figure formed by two rays sharing a common endpoint (vertex). In mathematical analysis and rotational geometry, angle measurement quantifies the degree of planar rotation required to superimpose one ray onto another. The SI coherent derived unit of angular displacement is the Radian ($ ext{rad}$), defined as the central angle subtended by an arc whose length is equal to the radius of the circle: $ heta = rac{s}{r}$.
Throughout the history of science, civil surveying, astronomy, and ballistics, several distinct angular systems became entrenched:
- Sexagesimal Degree System ($360^circ$): Developed in ancient Mesopotamia and Babylon, subdividing a full rotation into 360 equal parts, each degree further divided into 60 arcminutes ($'$) and 3,600 arcseconds ($''$).
- Circular Radian System ($2pi ext{ rad}$): The foundational unit of calculus and trigonometry. Radians simplify derivatives ($ rac{d}{dx}sin x = cos x$) because arc length formulas require no scalar conversion constants.
- Centesimal Gradian System ($400 ext{ grad}$): Standardized in France during the metric revolution, dividing a right angle into exactly 100 gradians (or gon), creating a circle of 400 grad.
- Milliradian / Mil System: Widely deployed in optical rifle reticles, laser rangefinding, and military artillery ballistics.
Angular Conversion Equations
Because a full rotation represents $360^circ$, $2pi ext{ radians}$, and $400 ext{ gradians}$, exact conversion factors apply:
- Degrees to Radians: $ ext{Radians} = ext{Degrees} imes rac{pi}{180}$
- Radians to Degrees: $ ext{Degrees} = ext{Radians} imes rac{180}{pi}$
- Degrees to Gradians: $ ext{Gradians} = ext{Degrees} imes rac{10}{9}$
- Degrees to Arcminutes & Arcseconds: $1^circ = 60' = 3,600''$
The Ballistic Mil-Relation Formula in Tactical Optics
In military marksmanship and artillery surveying, the milliradian (mrad / mil) provides a practical angular relationship where $1 ext{ mil}$ subtends an arc length of approximately $1 ext{ meter}$ at a range of $1,000 ext{ meters}$ ($1 ext{ cm}$ at $100 ext{ m}$):
$$ ext{Target Distance (meters)} = rac{ ext{Target Height (millimeters)}}{ ext{Subtension in Mils}}$$
This allows sniper spotters and artillery teams to calculate exact target ranges instantly in the field using simple mental arithmetic without needing laser telemetry devices.
Why is a full circle divided into 360 degrees?
The Babylonian base-60 (sexagesimal) numerical system selected 360 because 360 has 24 unique factors ($1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360$), allowing circles to be partitioned evenly into thirds, fourths, fifths, and sixths. It also closely corresponds to the ~365 days in an astronomical solar year.
What is the relationship between an arcminute and a nautical mile?
By historical definition, one nautical mile ($1,852 ext{ meters}$) was defined as exactly one arcminute ($1' = rac{1}{60}^circ$) of latitude along any meridian of Earth. Navigators can measure distance on nautical charts directly using latitude minute dividers.
What is the difference between true mils and NATO mils?
Mathematically, a true circle contains $2pi imes 1,000 pprox 6,283.185 ext{ milliradians}$. To simplify compass dials and artillery gear ratios, the NATO military alliance rounded this to exactly 6,400 mils per revolution, while the former Soviet Union rounded it to 6,000 mils.
Why do mathematical programming libraries require angles in radians?
Standard language math libraries (such as JavaScript Math.sin(), Python math.cos(), C sin()) require radians because trigonometric functions are computed via Taylor and Maclaurin power series expansions: $sin(x) = x - rac{x^3}{3!} + rac{x^5}{5!} - dots$, which are mathematically valid only when $x$ is expressed in radians.
While planar angles measure two-dimensional rotational divergence across a single plane, physical reality requires measuring angular spread across three dimensions. In the International System of Units, the SI derived unit of solid angle is the Steradian ($ ext{sr}$).
A solid angle ($Omega$) is defined as the surface area ($A$) of a sphere subtended by a cone originating from the sphere's center, divided by the square of the sphere's radius ($r$):
Because the total surface area of a sphere is $A = 4pi r^2$, a complete spherical shell subtends exactly $4pi ext{ steradians}$ ($pprox 12.56637 ext{ sr}$). Steradians are indispensable in optics, optical engineering, and photometry: Luminous Intensity (measured in Candelas, $ ext{cd}$) is defined as luminous flux per unit solid angle: $1 ext{ cd} = 1 ext{ lumen/steradian}$.
On the curved surface of Earth, standard Euclidean planar trigonometry breaks down. To calculate the shortest navigation flight path between two geographic coordinates on Earth's spherical surface (the Great-Circle Distance), navigators and GPS flight management systems execute the Haversine formula using angles converted strictly into radians:
$$a = sin^2left( rac{Delta phi}{2}
ight) + cos(phi_1) cdot cos(phi_2) cdot sin^2left( rac{Delta lambda}{2}
ight)$$
$$c = 2 cdot operatorname{atan2}left(sqrt{a}, sqrt{1-a}
ight)$$
$$d = R cdot c$$
Where $phi$ represents latitude in radians, $lambda$ represents longitude in radians, $R pprox 6,371 ext{ km}$ is the mean radius of Earth, and $d$ is the shortest spherical geodesic distance.
