Angle Converter & Sexagesimal DMS Calculator

Convert planar angle measurements across Degrees, Radians, Gradians, Arcsec, NATO Mils, and Turns with live Sexagesimal DMS coordinates & subtense rangefinding.

Geometric, Military & Astronomical Milestones
🎯 Estimate-First Geometric Challenge

Before checking below, can you estimate how many Radians (rad) equal 180 °?

Target (Radians)
3.14159265
Sexagesimal DMS & Optical Subtense (W ≈ D · θ)
180° 0' 0"
Width W = 3,141.593 m @ 1000 m Range
Canonical geodetic coordinates & military subtense rangefinding
Simultaneous 14-Unit Equivalency Matrix
Degrees (°)
180
Radians (rad)
3.141593
Gradians (gon) (grad)
200
Arcminutes (')
10,800
Arcseconds (")
648,000
Milliarcseconds (mas)
6.480000e+8
Milliradians (SI) (mrad)
3,141.592654
NATO Artillery Mils (1/6400) (mil (NATO))
3,200
Soviet / Warsaw Mils (1/6000) (mil (USSR))
3,000
Swedish Mils (1/6300) (mil (SE))
3,150
Turns / Revolutions (turn)
0.5
Quadrants (Right Angle) (quad)
2
Sextants (60°) (sextant)
3
Zodiac Signs (30°) (sign)
6
Step-by-Step Mathematical Walkthrough
Step 1: Convert source unit to base SI Radians (rad)
Radians = 180 ° × 0.0174532925199 = 3.14159265 rad
Step 2: Convert Radians to target unit (Radians)
Radians = 3.14159265 rad / 1.00000000000 = 3.14159265 rad
Step 3: Sexagesimal DMS (Degrees° Minutes' Seconds") Decomposition
Decimal Degrees = 180.000000° → Integer D = 180°, Minutes M = 0', Seconds S = 0.0000"
Step 4: Optical Subtense Target Width (W ≈ D · θ_rad)
At Range D = 1,000 m and Angle θ = 3.141593 rad → Subtended Target Width W = 3,141.5927 m

About the Angle Converter

An angle converter is an essential tool in mathematics, geometry, physics, and mechanical engineering used to translate angular measurements between different units of circle division. Angles represent the amount of rotation between two intersecting lines and are fundamental to trigonometry, astronomy, navigation, and computer graphics calculations.

Mathematical Formula & Logic

The core mathematical conversions rely on the relation that a full circle (1 turn) is equivalent to 360 degrees, 2π radians, or 400 gradians. - Degrees to Radians: Rad = Deg × (π / 180) - Radians to Degrees: Deg = Rad × (180 / π) - Degrees to Gradians: Grad = Deg × (400 / 360) - Degrees to Turns: Turn = Deg / 360

Step-by-Step Example

Convert 90 degrees to Radians: 90° × (π / 180) = 90 × 0.01745329 = 1.570796 radians (which is exactly π / 2). The error this converter exists to prevent Almost every wrong trigonometry answer comes from one cause: the calculator was in the wrong angle mode. The failure is silent, because both answers are perfectly valid numbers. sin(90°) = 1.000000, the maximum value the sine function reaches. sin(90 radians) = 0.893997, because 90 radians is a little over fourteen full turns and lands somewhere unremarkable. Nothing warns you. If a trigonometric result looks plausible but disagrees with the textbook, check the mode before checking the algebra. Why radians are the natural unit Degrees are a human convention; radians are the ratio the geometry itself produces. One radian is the angle that sweeps an arc equal in length to the radius, which makes arc length simply s = r × θ. A radius of 5 through 1.2 radians traces exactly 6 units of arc. Attempt that in degrees and a conversion factor has to be smuggled in. The same property makes calculus behave. The derivative of sin(x) is cos(x) only when x is in radians; in degrees it picks up a stray factor of π/180 that propagates through every subsequent step. It is also why the small angle approximation works: for small θ in radians, sin(θ) ≈ θ. At θ = 0.01 the sine is 0.009999833, an error under two parts in ten million. Physics leans on this constantly, from pendulums to optics. Dividing the circle Every angular unit is defined by how many of it fit in one full turn, so conversion is always a matter of rescaling that count: 360 degrees. 400 gradians. 21,600 arcminutes. 1,296,000 arcseconds. 2π radians. The finer divisions belong to disciplines that need them. Astronomy works in arcseconds because stellar parallax is measured in fractions of one. Navigation and surveying use degrees, minutes and seconds, where 40° 26′ 46″ means 40 degrees plus 26 sixtieths plus 46 thirty-six-hundredths. A trap worth knowing: military mils are not one unit but three. NATO divides the circle into 6,400 mils, the former Soviet system into 6,000, and Sweden into 6,300. They exist because a mil subtends roughly one metre at a kilometre, which makes artillery correction arithmetic trivial — but a figure converted with the wrong national definition is out by up to 6%.

Reference Data & Values

unitdegradgrad
1 Turn (Full Circle)360°2π rad (~6.283)400 grad
Half Circle180°π rad (~3.142)200 grad
Right Angle90°π/2 rad (~1.571)100 grad
Standard Angle45°π/4 rad (~0.785)50 grad
Sixth of a Turn60°π/3 rad (~1.047198)66.6667 grad
Twelfth of a Turn30°π/6 rad (~0.523599)33.3333 grad
One Radian57.295780°1 rad63.6620 grad
One Arcminute0.016667°0.000290888 rad0.018519 grad
One NATO Mil0.05625°0.000981748 rad0.0625 grad

Frequently Asked Questions

A radian is the standard unit of angular measure defined by the arc length of a circle equal to its radius. It is preferred in calculus because it simplifies trigonometric derivatives and integrals, avoiding complex conversion factors.
The 360-degree circle originated from ancient Babylonian astronomers. They used a sexagesimal (base-60) numerical system and tracked the sun's movement, which was approximately 1 degree per day across a 360-day calendar cycle.
A gradian (or gon) is a unit of angle equal to 1/400 of a full circle. It was designed in France to metricate angular measurement, making right angles exactly 100 gradians for easier decimal calculations.
Almost always because it is in the wrong angle mode, and the failure is silent since both results are valid numbers. sin(90°) is exactly 1, while sin(90 radians) is 0.893997 — 90 radians being a little over fourteen complete turns. If a trigonometric result looks plausible but disagrees with the expected value, check DEG against RAD before checking your algebra.
For small angles measured in radians, sin(θ) is very close to θ itself. At θ = 0.01 radians the sine is 0.009999833, an error below two parts in ten million. The approximation only holds in radians, and it is what makes pendulum motion, lens optics and many vibration problems tractable — replacing the sine with the angle turns an awkward equation into a linear one.
A mil is an angle subtending roughly one metre at one kilometre, which makes artillery correction arithmetic simple. The problem is that no single definition won: NATO divides the circle into 6,400 mils, the former Soviet system into 6,000, and Sweden into 6,300. Converting with the wrong national definition puts the result out by up to 6%, so the standard has to be stated rather than assumed.
The notation subdivides sexagesimally, exactly like time. One degree is 60 arcminutes, written with a prime, and one arcminute is 60 arcseconds, written with a double prime. So 40° 26′ 46″ is 40 + 26/60 + 46/3600 = 40.4461 degrees. Navigation, surveying and astronomy still use it because a full circle contains 1,296,000 arcseconds, giving very fine resolution without decimals.