Angle Converter

Convert between SI and historical plane-angle units (radian, milliradian, degree, gradian, turn, arcminute, arcsecond) anchored to the BIPM SI Brochure definition of the radian.

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From

Value

Result

0.0174533 rad

1 °≈ 0.017453 rad

Decimals

All units

Unit	Value
Radian (rad)	0.0174533
Milliradian (mrad)	17.4533
Degree (°)	1
Gradian (gon)	1.11111
Turn (tr)	0.00277778
Arcminute (arcmin)	60
Arcsecond (arcsec)	3,600

آخر تحديث: 20 مايو 2026

Angle converter. SI and historical plane-angle units with BIPM-anchored definitions.

An angle converter switches a value between radians, degrees, gradians, turns, arcminutes, arcseconds and milliradians using BIPM-anchored definitions. It tags each result exact or approximate so trigonometry students, surveyors, astronomers and rifle shooters can tell finite-decimal conversions from π-derived ones rounded by IEEE-754 doubles.

What Is an Angle Converter?

An angle converter is a tool that takes a plane-angle measurement in one unit and returns the equivalent value in any other supported unit. It works by routing every conversion through a single base unit, the radian, using each unit's defined factor: 1 milliradian is exactly 10⁻³ rad, 1 turn is 2π rad, 1 degree is π/180 rad, 1 gradian is π/200 rad, 1 arcminute is π/10,800 rad, and 1 arcsecond is π/648,000 rad. The radian itself is the SI coherent derived unit for plane angle, defined in BIPM SI Brochure §2.3.4 as the angle subtended at the centre of a circle by an arc equal in length to the radius. The same plane-angle definitions appear in ISO 80000-3:2019 (Quantities and units, Part 3 — see the linked standard in Sources at page bottom).

This converter supports seven units across three groups. The SI group covers the radian and the milliradian (1 mrad = 10⁻³ rad), both used in physics and engineering. The common group covers the degree (Babylonian sexagesimal, used in everyday geometry, CAD, navigation and education), the gradian or gon (post-1793 French metric, still used in continental European cadastral surveying — 400 gon = 1 turn means a right angle is a clean 100 gon), and the turn (one full rotation = 2π rad = 360° = 400 gon, used in rotational mechanics and motor specifications). The scientific group covers the arcminute and the arcsecond, the sexagesimal subdivisions used in astronomy and precision navigation: the Hubble Space Telescope resolves about 0.05 arcseconds, the James Webb Space Telescope's NIRCam reaches about 0.07 arcseconds at 2 µm (diffraction-limited), and a healthy human eye barely resolves one arcminute at reading distance.

What distinguishes a serious angle converter from a marketing widget is honesty about precision. The factor 1 rad ≈ 57.2958° is not exact; it is a six-significant-figure rounding of 180/π = 57.29577951308232…, which is irrational because π is irrational. The exact relationship runs the other way: 1° = π/180 rad by definition. This converter labels rad↔mrad with an "exact" badge because both units are tagged exact in the source data, but it labels every conversion that touches degree, gradian, turn, arcminute or arcsecond as approximate — not because the underlying mathematical definitions are fuzzy (they are bit-exact in symbolic form), but because π cannot be represented exactly as an IEEE-754 double, so any π-derived factor in 64-bit floating point carries a few ulps of representational drift. The badge tells you the truth: degree↔radian is definitionally exact, computationally approximate.

How to Convert Between Angle Units

Every angle conversion is one multiplication and one division through the radian. The general formula is:

y = x \cdot \frac{a_{\text{from}}}{a_{\text{to}}}

where $x$ is your input value, $a_\text{from}$ is the source unit's factor to radians, and $a_\text{to}$ is the target unit's factor to radians. To do it by hand:

1. Look up the source-to-radian factor. For degrees, $a_\text{from} = \pi/180 \approx 0.01745329$.

2. Multiply the input by that factor to get radians. 90° × π/180 = π/2 rad ≈ 1.5707963 rad.

3. Look up the target-to-radian factor. For gradians, $a_\text{to} = \pi/200 \approx 0.01570796$.

4. Divide the radian value by the target factor. (π/2) ÷ (π/200) = 100 gon (the π cancels, giving the exact rational answer).

The same procedure works for every supported unit. Going from arcseconds to radians: 1″ × π/648,000 ≈ 4.848137 × 10⁻⁶ rad. Going from turns to degrees: 0.25 tr × 2π rad/tr ÷ (π/180 rad/°) = 90° (again π cancels). Going from milliradians to MOA (arcminutes): 1 mrad × 10⁻³ rad/mrad ÷ (π/10,800 rad/′) = 10,800 / (1000 π) ≈ 3.4377 MOA — the canonical ballistic conversion.

To use this calculator, pick the source unit from the "From" dropdown, type a value, then pick the target unit from the "To" dropdown. The result updates on every keystroke. Click the result card to copy it to the clipboard. Use the precision selector to switch between auto (6 significant figures), or a fixed 0, 2, 4, 6, 10 or 15 decimals. Auto-precision switches to scientific notation when the result is larger than 10¹² (one trillion) or smaller than 10⁻³, so converting an astronomical arcsecond to turns still renders readably. The "exact" badge appears only for the rad↔mrad pair, because that is the single conversion in this set that does not pass through π in the IEEE-754 double representation.

Angle Conversion Formula

y = x \cdot \frac{a_{\text{from}}}{a_{\text{to}}}

$y$ = The converted value, expressed in the target unit of plane angle.
$x$ = The input value, expressed in the source unit of plane angle.
$a_{\text{from}}$ = Factor that converts the source unit to radians (e.g. π/180 for degrees, π/200 for gradians, 2π for turns, 10⁻³ for milliradians).
$a_{\text{to}}$ = Factor that converts the target unit to radians (e.g. π/10800 for arcminutes, π/648000 for arcseconds).

The formula is a two-step pivot through the radian, the SI coherent derived unit for plane angle (BIPM SI Brochure §2.3.4). The factor table this calculator uses is sourced from the BIPM SI Brochure (the matching ISO 80000-3 standard is linked at page bottom):

Radian (rad): 1 rad (SI coherent derived unit, exact)
Milliradian (mrad): 10⁻³ rad (SI prefix, exact)
Degree (°): π/180 rad ≈ 0.0174532925199433 rad (exact definition; representational drift in IEEE-754)
Gradian (gon): π/200 rad ≈ 0.0157079632679490 rad (exact definition; 400 gon = 1 turn)
Turn (tr): 2π rad ≈ 6.283185307179586 rad (exact definition; 1 tr = 360° = 400 gon)
Arcminute (′, arcmin): π/10,800 rad ≈ 2.908882086657216 × 10⁻⁴ rad (= 1°/60, exact definition)
Arcsecond (″, arcsec): π/648,000 rad ≈ 4.848136811095360 × 10⁻⁶ rad (= 1°/3600, exact definition)

For the degree → radian direction the calculator computes 1° × π/180 ≈ 0.0174532925199433 rad. The result is mathematically exact (1° = π/180 rad by definition) but stored as a 64-bit double, so the last few digits are a rounding of the irrational π. The reverse direction 1 rad → degrees gives 180/π ≈ 57.2957795130823°, again irrational because π is. Conversions that stay inside the SI group (rad ↔ mrad) are bit-exact and earn the "exact" badge; everything else is tagged approximate as a true label of the IEEE-754 representation, not the mathematical definition.

Worked Angle Conversion Examples

90 degrees to radians (canonical trigonometry)

Set From = Degree, To = Radian, Value = 90. The formula gives 90 × π/180 = π/2 rad ≈ 1.5707963 rad at auto-precision. Set precision to 15 decimals to see 1.570796326794897 rad — the full IEEE-754 mantissa for π/2. This is the conversion every trigonometry student does first: sin(π/2) = 1 in radians is the same statement as sin(90°) = 1 in degrees, but only the radian version makes the Taylor series and derivative identities work without an extra π/180 factor in every line. The result is tagged approximate because π/2 is irrational and any 64-bit double is a rounding.

1 mrad to MOA (rifle scopes, ballistic shooters)

Set From = Milliradian, To = Arcminute, Value = 1. The formula gives 1 mrad × 10⁻³ rad/mrad ÷ (π/10,800 rad/′) = 10,800 / (1000π) ≈ 3.4377 arcmin (MOA). At 100 yards, 1 MOA covers roughly 1.047 inches and 1 mrad covers about 3.6 inches — so 1 mrad ≈ 3.44 MOA worth of point-of-aim shift. This is why long-range shooters mix the two: a mrad-marked reticle reads 0.1 mrad per click (typical), which equals about 0.344 MOA per click, finer than the common ¼-MOA-per-click scopes used in hunting. Both endpoints involve π, so the conversion is tagged approximate even though the underlying definitions are exact.

1 turn to degrees, gradians, and radians

Set From = Turn, To = Degree, Value = 1. The formula gives 1 tr × 2π rad/tr ÷ (π/180 rad/°) = 360° (π cancels). Switching To = Gradian gives 2π / (π/200) = 400 gon. Switching To = Radian gives 2π ≈ 6.2831853 rad. The first two conversions are exact rationals (the irrational π cancels in the ratio); the third is irrational and tagged approximate. This is the textbook proof that 1 full rotation = 360° = 400 gon = 2π rad — the same angle measured three ways. Servo motors and rotary encoders are usually specified in turns or fractions of a turn; CAD software speaks degrees; physics speaks radians.

1 arcsecond to radians (Hubble-scale astronomy)

Set From = Arcsecond, To = Radian, Value = 1. The formula gives 1″ × π/648,000 ≈ 4.84814 × 10⁻⁶ rad (≈ 4.85 μrad). Auto-precision switches to scientific notation because the result is below 10⁻³. The Hubble Space Telescope resolves about 0.05 arcseconds, or 2.42 × 10⁻⁷ rad; the James Webb Space Telescope's NIRCam reaches about 0.07 arcseconds at 2 µm (diffraction-limited). By comparison, the naked human eye barely resolves one arcminute (≈ 2.91 × 10⁻⁴ rad). Stellar parallax — the basis of the parsec — uses arcseconds as the angular base: a parsec is the distance at which 1 AU subtends 1 arcsecond.

100 gon to degrees (French cadastral survey)

Set From = Gradian, To = Degree, Value = 100. The formula gives 100 × (π/200) ÷ (π/180) = 100 × 180/200 = 90° (π cancels, exact rational answer). A right angle in the centesimal system is a clean 100 gon — that is exactly why post-revolutionary France invented the unit in 1793. French and Swiss cadastral surveys, topographic maps and theodolites built before the 1990s commonly read in gons because a quadrant divides into 100 equal parts instead of 90, making decimal subdivisions cleaner. The European Union legally recognises the gon for surveying applications; the BIPM SI Brochure does not list it among the SI-accepted units.

Comparative table: 1 unit in radians

Unit	Symbol	Value in radians	Exact?
Radian	rad	1	yes
Milliradian	mrad	1 × 10⁻³	yes
Arcsecond	″	π/648,000 ≈ 4.848 × 10⁻⁶	def. exact, IEEE-754 rounded
Arcminute	′	π/10,800 ≈ 2.909 × 10⁻⁴	def. exact, IEEE-754 rounded
Gradian	gon	π/200 ≈ 0.01571	def. exact, IEEE-754 rounded
Degree	°	π/180 ≈ 0.01745	def. exact, IEEE-754 rounded
Turn	tr	2π ≈ 6.28319	def. exact, IEEE-754 rounded

Use this table to do conversions by hand: divide source-in-radians by target-in-radians. For example, 1 turn in arcseconds = 2π / (π/648,000) = 2 × 648,000 = 1,296,000″ (a full circle has exactly 1,296,000 arcseconds, an exact integer because the π cancels).

Most common angle conversions

These are the unit pairs people look up most often. Use them as a quick reference, or paste them into the calculator for an exact result to 15 decimals.

Degrees to radians: 1° ≈ 0.017453 rad (exact: π/180)
Radians to degrees: 1 rad ≈ 57.29578° (exact: 180/π)
Turn to radians: 1 tr = 2π rad ≈ 6.28319 rad
Turn to degrees: 1 tr = 360° (exact)
Turn to gradians: 1 tr = 400 gon (exact)
Degrees to gradians: 1° = 10/9 gon ≈ 1.11111 gon (exact rational)
Gradians to degrees: 1 gon = 9/10° = 0.9° (exact)
Degree to arcminutes: 1° = 60′ (exact, by definition)
Arcminute to arcseconds: 1′ = 60″ (exact)
Degree to arcseconds: 1° = 3,600″ (exact)
Milliradians to MOA (arcmin): 1 mrad ≈ 3.43775 arcmin
MOA to milliradians: 1 arcmin ≈ 0.29089 mrad
Arcsecond to microradians: 1″ ≈ 4.84814 μrad
Full circle to arcseconds: 1 tr = 1,296,000″ (exact integer)

Angle Conversion Tips

Pivot through the radian. Every conversion in this tool is implemented as "input × from-factor ÷ to-factor", with the radian as the pivot. Memorising six factors gives you every cross-conversion for free: degree (π/180), gradian (π/200), turn (2π), arcminute (π/10,800), arcsecond (π/648,000), and milliradian (10⁻³).
Read the "exact" badge as a statement about IEEE-754, not mathematics. 1° = π/180 rad is mathematically exact by definition. The badge only fires on rad ↔ mrad because that is the single pair in this set that does not multiply or divide by π in 64-bit floating point. Every degree, gradian, turn, arcminute and arcsecond conversion is tagged approximate to honestly report the few-ulp representational drift, not because the underlying definition is fuzzy.
For rifle scopes, remember the mil-vs-MOA conversion. 1 mrad ≈ 3.4377 arcmin (MOA), and 1 arcmin ≈ 0.2909 mrad. At 100 yards, 1 MOA covers about 1.047 inches; 1 mrad covers about 3.6 inches. Most mrad-reticle scopes have 0.1-mrad clicks; most MOA scopes have ¼-MOA clicks. The mrad click is finer (≈ 0.344 MOA equivalent) — useful at extended range, less convenient at close range where the click is sub-inch.
Use radians for any calculus or physics formula. The derivative of sin(x) equals cos(x) only when x is in radians; in degrees you pick up a π/180 factor that ruins Taylor series and most differential equations. CAD software typically accepts degrees in the UI but converts internally to radians for the trigonometric kernels — which is why entering 0.523599 (≈ π/6) gives the same result as entering 30°.
Use degrees-minutes-seconds (DMS) for navigation and surveying, decimal degrees for software. Charts and theodolites still print 60° 35′ 15″, but every GIS package wants the decimal form 60.5875°. The conversion is straightforward: decimal degrees = degrees + minutes/60 + seconds/3600. This calculator does not parse DMS strings directly — convert each part to its decimal-degree equivalent and add them, or paste each arcmin/arcsec value separately.
Watch out for NATO mil vs true milliradian. A true milliradian is exactly 1/1000 rad and gives 2000π ≈ 6283.19 mrad per full circle. The NATO mil rounds to 6400 per circle to make artillery subdivisions cleaner, so 1 NATO mil ≈ 0.05625° ≠ 1 true mrad. This calculator implements the true milliradian; if your scope reticle is NATO-mil-marked, multiply by 6400/(2000π) ≈ 1.01859 before pasting in.
Use turns for rotation specs and unit-circle reasoning. "1.5 turns" is far more readable than "3π rad" or "540°" when you are talking about a winch, a stepper motor, or a knob — and it is what most motor datasheets quote. The turn is also the cleanest way to teach the unit circle: angles between 0 and 1 turn map directly to fractions of the circle, no awkward 0–360 or 0–2π scaling needed.
When the result is bigger than 10¹² (one trillion) or smaller than 10⁻³ at auto-precision, the display switches to scientific notation. This is on purpose: converting a single arcsecond to turns gives ≈ 7.72 × 10⁻⁷ tr — readable in scientific notation, unreadable as 0.000000772 tr.

Angle Converter — Frequently Asked Questions

Is this angle converter free?

Yes. The calculator is free, requires no account, runs entirely in your browser, and is ad-free. The embeddable iframe version at /widget/angle-converter is also free and ad-free, so you can drop it into trigonometry course materials, ballistic blogs, surveying portals or astronomy clubs without exposing readers to third-party trackers.

How accurate are the conversion factors?

The radian and milliradian factors are exact in IEEE-754: 1 mrad = 10⁻³ rad exactly. The degree (π/180), gradian (π/200), turn (2π), arcminute (π/10,800) and arcsecond (π/648,000) are mathematically exact by definition — the BIPM SI Brochure prints these as bit-exact ratios — but every conversion involving them carries a few ulps of representational drift because π cannot be represented exactly as a 64-bit double. The calculator labels those conversions approximate to be honest with you.

How do I convert degrees to radians?

Multiply degrees by π/180. So 30° = 30 × π/180 = π/6 ≈ 0.5236 rad, 45° = π/4 ≈ 0.7854 rad, 90° = π/2 ≈ 1.5708 rad, and 180° = π ≈ 3.1416 rad. The relationship is exact by definition; the decimal value is irrational because π is irrational. In this calculator, set From = Degree, To = Radian, then type the degree value — the result updates on every keystroke.

How do I convert radians to degrees?

Multiply radians by 180/π. So 1 rad ≈ 57.2958°, π/2 rad = 90° exactly, π rad = 180° exactly, and 2π rad = 360° exactly. The reverse direction is irrational in form: 180/π = 57.29577951308232… is an irrational number you can print to as many decimals as you want, but never finitely. For 15 decimals, set the precision selector to 15.

What is 1 mrad in MOA?

One milliradian equals approximately 3.4377 minutes of angle (MOA). The exact formula is 10,800 / (1000π). At 100 yards, 1 mrad covers about 3.6 inches; 1 MOA covers about 1.047 inches.

What is the difference between a gradian and a degree?

A gradian (gon) is 1/400 of a full circle; a degree is 1/360. 1 gon = 0.9° exactly, and 1° = 10/9 gon ≈ 1.111 gon. A right angle is 100 gon vs 90°. The gradian was invented in post-revolutionary France in 1793 to give a decimal subdivision of the right angle; it is still standard in French and Swiss cadastral surveying, but it is not part of the SI and rarely appears outside surveying applications.

What is an arcsecond and where is it used?

An arcsecond is 1/3,600 of a degree, or π/648,000 rad ≈ 4.848 × 10⁻⁶ rad. It is the standard angular unit in observational astronomy: the Hubble Space Telescope resolves about 0.05 arcseconds, the James Webb Space Telescope's NIRCam reaches about 0.07 arcseconds at 2 µm, and stellar parallax measurements (which define the parsec) work in arcseconds. The human eye barely resolves one arcminute, so any arcsecond-scale detail requires a telescope.

Why does 1 turn equal 2π radians?

Because the radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. The unit circle has circumference 2π, so a full rotation sweeps exactly 2π such arcs.

What is the difference between true milliradian and NATO mil?

A true milliradian is exactly 10⁻³ rad, giving 2000π ≈ 6,283.19 mrad per full circle. The NATO mil rounds to 6,400 per circle for cleaner artillery subdivisions, so 1 NATO mil ≈ 0.05625° ≠ 1 true mrad — the two units differ by about 1.86%. The Warsaw Pact mil used 6,000 per circle, and the Swedish streck used 6,300. This calculator implements the true milliradian; multiply by 6400/(2000π) ≈ 1.01859 if your reticle is NATO-mil-marked.

How do I convert degrees-minutes-seconds (DMS) to decimal degrees?

Use the formula: decimal degrees = degrees + minutes/60 + seconds/3,600. For example, 60° 35′ 15″ = 60 + 35/60 + 15/3,600 = 60.5875°. Then enter that decimal-degree value in this calculator. Charts and surveying instruments still print DMS, but every GIS package and most calculators expect the decimal form.

Can I embed this converter on my site?

Yes. The embeddable version lives at /widget/angle-converter; copy the iframe snippet from the embed page. The iframe is ad-free, dependency-free, mobile-responsive, and inherits no third-party trackers — useful for trigonometry teachers, surveying-firm intranets, ballistic blogs and astronomy clubs that want a clean angle tool without sending readers to a third-party page.

Is the gradian the same as the gon?

Yes. "Gradian," "gon," and "grade" all name the same unit: 1/400 of a full circle, exactly π/200 radians, exactly 0.9 degrees. The international standard symbol is gon (ISO 80000-3).

Key Angle Conversion Terms

Radian (rad)

The SI coherent derived unit for plane angle, defined in BIPM SI Brochure §2.3.4 as the angle subtended at the centre of a circle by an arc equal in length to the radius. One full turn equals 2π radians; 1° = π/180 rad. The radian is dimensionless and is the natural unit for calculus, physics, and signal processing.

Milliradian (mrad)

One thousandth of a radian, exact by SI prefix. Used in rifle scopes (0.1-mrad scope clicks), artillery sights, and engineering tolerancing. 1 mrad ≈ 3.4377 arcminutes (MOA); 1 mrad covers about 1 cm at 100 m or about 3.6 inches at 100 yards.

Degree (°)

An SI-accepted non-SI unit equal to π/180 radians or 1/360 of a turn. Babylonian sexagesimal origin (60-divisible number, useful for divisor-rich arithmetic). Standard in everyday geometry, CAD/CNC, navigation, latitude/longitude, and most engineering drawings. Codified in ISO 80000-3:2019.

Gradian (gon, grade)

A unit of plane angle equal to 1/400 of a turn, exactly π/200 radians or 0.9 degrees. Invented in post-revolutionary France in 1793 as a decimal subdivision of the right angle (100 gon = 90°). Still standard in continental European cadastral surveying. Not part of the SI; ISO 80000-3 specifies the symbol "gon."

Turn (tr)

One full rotation: exactly 2π radians, 360 degrees, or 400 gradians. Used in rotational mechanics, motor and encoder specifications, and unit-circle pedagogy. Symbols: tr, rev, cyc. Also called a cycle, revolution or full circle.

Arcminute (′, arcmin, MOA)

One sixtieth of a degree, exactly π/10,800 radians. In astronomy and navigation it is the angular unit at which the human eye barely resolves detail (≈ 1 arcmin acuity). In firearms it is called "minute of angle" (MOA) and equals about 1.047 inches at 100 yards. The full Moon subtends about 31 arcminutes.

Arcsecond (″, arcsec)

One sixtieth of an arcminute, or 1/3,600 of a degree — exactly π/648,000 radians ≈ 4.848 × 10⁻⁶ rad. The standard unit in observational astronomy: Hubble resolves about 0.05 arcseconds, JWST's NIRCam about 0.07 arcseconds at 2 µm. A parsec is the distance at which 1 AU subtends exactly 1 arcsecond.

BIPM SI Brochure

The defining text of the International System of Units, maintained by the Bureau International des Poids et Mesures. The 9th edition (2019, with corrections to 2025) defines the radian in §2.3.4 as the SI coherent derived unit for plane angle. ISO 80000-3:2019 mirrors the same definitions for plane-angle units.

Significant figures

The digits in a numerical result that carry meaning about its precision. Auto-precision in this calculator targets 6 significant figures, the default for engineering tables. NIST SP 811 quotes conversion factors to 6 or 7 significant figures.

IEEE-754 double

The 64-bit floating-point format used by JavaScript and most calculators. It carries about 15–17 significant decimal digits. π cannot be represented exactly in this format, so every angle conversion that multiplies or divides by π carries a few ulps of representational drift — which is why the calculator labels degree/gradian/turn/arcmin/arcsec conversions "approximate" even though the mathematical definitions are exact.

Scientific notation

A way of writing very large or very small numbers as a coefficient times a power of ten, e.g. 1 arcsecond = 4.84814 × 10⁻⁶ rad. The calculator switches to this notation automatically at auto-precision when results fall outside the 10⁻³ to 10¹² range.

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