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Angle Converter

1Degree = 0.017453Radian
0.017453

Angle System

Arcsecond (″)3600
Arcminute (′)60
Degree (°)1
Right Angle0.011111
Circle0.002778

Radian System

Radian (rad)0.017453
Milliradian (mrad)17.453293

Free online angle converter covering 7 commonly used units: degree (°), arcminute ('), arcsecond ("), right angle, turn (revolution), radian (rad), milliradian (mrad). Enter a value, pick any two units, and get instant bidirectional results. All calculations run locally in your browser — no data is uploaded.

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About Angles and Angle Units

An angle is a quantity used to measure rotation or the amount of opening between two rays. In everyday life, the most familiar system is degrees: dividing a full circle into 360 equal parts, each 1 part being 1 degree (°). This division originated from the ancient Babylonian sexagesimal system, and because 360 is easily divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, etc., it has persisted to this day in surveying, navigation, engineering, and everyday expression.

The radian (rad) is the standard unit of angle in the International System of Units (SI). Its definition is: when the length of an arc equals the radius, the central angle subtended by that arc is 1 radian. Since the circumference is 2πr, 1 turn = 2π rad ≈ 6.2832 rad. Radians make calculus, trigonometric functions, and physics formulas the most concise; for example, lim(x→0) sin x/x = 1 holds only when x is in radians.

Arcminutes (') and arcseconds (") are subdivisions of the degree system: 1° = 60', 1' = 60", so 1° = 3600". Milliradians (mrad) are thousandths of a radian: 1 mrad = 0.001 rad ≈ 0.0573°. Arcminutes/arcseconds are common in astronomy, geographic coordinates, and precision measurement; milliradians are common in ballistics, optics, laser ranging, and engineering small-angle estimation.

Right angles and turns are two more intuitive angle units: 1 right angle = 90°, 1 turn (revolution) = 360° = 2π rad. Right angles are suitable for describing perpendicular relationships in geometry; turns are suitable for describing rotation counts, such as motor RPM (revolutions per minute), which is essentially the number of turns rotated per unit time.

Why distinguish between degrees and radians? In mathematics and programming, trigonometric functions almost always default to radian input. If you pass 90 directly as degrees to Python's math.sin, it computes sin(90 rad), which is approximately 0.894, not the expected 1. Many beginners' first bug comes from here. This tool puts both units side by side so you can quickly verify before entering code or formulas.

All conversions on this page are pure linear proportional relationships with no offset like temperature. The core formulas are just two: rad = deg × π/180, deg = rad × 180/π. The remaining units (arcminute, arcsecond, right angle, turn, milliradian) are all derived from these two relationships, so conversion results are precise and reproducible.

Use Cases

  • Students converting between degrees and radians to verify answers when doing trigonometry, analytic geometry, and calculus homework
  • Programmers writing trigonometric functions in Python / JavaScript / C++, converting degree inputs to radians before calling sin/cos/tan
  • Engineers reading angle annotations on mechanical drawings or CAD models, converting degrees-minutes-seconds to decimal degrees or radians
  • Astronomy enthusiasts using telescope equatorial mounts or star chart software, converting right ascension/declination DMS to/from radians
  • Nautical, aviation, and surveying personnel working with azimuth/heading angles, switching between degrees, DMS, and turns
  • Game developers handling character facing, camera rotation, quaternions and Euler angles, converting between radians and degrees
  • 3D modelers and animators in Blender, Maya, etc., aligning degree rotation parameters with radian values in scripts
  • Optics, laser, and shooting fields using small-angle approximation, quickly cross-referencing milliradians (mrad) with arcminutes and degrees
  • Motor, encoder, and robotics engineers converting revolutions (rev), turns to/from radians and degrees
  • GIS professionals converting DMS-formatted latitude/longitude to decimal degrees for GIS software import
  • Teachers creating courseware or exam papers, generating degree-radian conversion tables in bulk to help students understand both units
  • DIY woodworking, metalworking, sewing, converting angle annotations on drawings to units readable by on-site protractors
  • Drone pilots reading gimbal pitch/heading angles, switching between degrees displayed in apps and radian values in mission planning software
  • Physicists writing papers, uniformly converting experimental angle records to SI radians to ensure formula consistency
  • Cross-border e-commerce or foreign trade personnel checking angle parameters in product specifications, unifying units commonly used in different countries
  • Using "Expand Conversion Results" to see all results for the same value across 7 angle units at once for cross-comparison

How to Use

  1. Enter a value in the left input box (default is 1)
  2. Select the source unit from the left dropdown and the target unit from the right dropdown; results are displayed instantly
  3. Click the swap button between the two dropdowns to flip source/target units with one click and view the reverse conversion
  4. Click "Expand Conversion Results" to view results for that value across all 7 angle units at once

Features

  • Full coverage of 7 angle units: degree (°), arcminute ('), arcsecond ("), right angle, turn (revolution), radian (rad), milliradian (mrad), covering both sexagesimal and radian systems
  • Instant bidirectional conversion between any two units: switching source/target units refreshes results in real time without re-entering values
  • One-click expand all results: view conversions of the input value across all 7 units on the same page, grouped by sexagesimal/radian system
  • Swap button in the middle: single click to flip source and target units for reverse conversion without retyping
  • Scientific notation for very small values: extremely small units like arcseconds and milliradians automatically use friendly formatting for better readability
  • Pure browser-local calculation: based on fixed conversion factors using multiplication/division, no network requests, data never leaves your device
  • Adapted for trigonometry/programming/engineering/navigation scenarios: clearly distinguishes radians (mathematical default) from degrees (commonly used in everyday measurement)

Degrees vs Radians: When to Use Which?

Neither angle system is absolutely better; the key is the scenario and what the tool defaults to. The table below helps you decide quickly.

Comparison ItemDegrees (° / ' / ")Radians (rad / mrad)
DefinitionCircle divided into 360 equal parts, each 1°Central angle when arc length equals radius
Best forEngineering surveying, map navigation, astronomical coordinates, everyday useMathematical analysis, programming trig functions, physics formulas, calculus
Calculation convenienceIntuitive, but trig derivatives/integrals require extra coefficientsMakes formulas like sin'x = cosx the most concise, no extra coefficients
Programming defaultMost languages require explicit conversion; otherwise interpreted as radiansDefault input for Python / JS / C++ / Java, etc.
SubunitsArcminutes ('), arcseconds ("), 1° = 3600"Milliradians (mrad), 1 rad = 1000 mrad

Best Practices

Before calling trig functions in code, confirm the unit is radians

The vast majority of programming languages (Math.sin / math.cos / np.sin, etc.) accept radians by default. If you have degrees, first convert to radians using this tool or via code (deg × π/180), otherwise results will be wrong. For example, 90° must first be converted to π/2 rad.

For engineering drawings, prioritize decimal degrees, then convert to radians as needed

Drawings commonly use DMS format like 30°15'20"; use this tool to first unify to 30.2556° or 0.5281 rad before substituting into calculations, to avoid DMS conversion errors.

Prioritize milliradians (mrad) for small-angle estimation

When the angle is less than 5°, 1 mrad ≈ 0.0573°, and sin θ ≈ θ (in radians). At 100 m distance, 1 mrad corresponds to approximately 10 cm, making it very suitable for quick mental calculations in shooting, laser, and optics.

Distinguish between "turn" and "revolutions per minute (RPM)"

A turn is an angle unit equal to 360°; RPM is a rotational speed unit representing how many turns per minute. The two are related but dimensionally different; do not treat 1 turn directly as 1 RPM.

When converting DMS geographic coordinates to decimal degrees, divide by the correct bases

Divide minutes by 60, seconds by 3600, then add to degrees. A common mistake is dividing seconds by 60 as well, causing coordinate offsets of tens of meters.

Pay attention to arcsecond-level precision in astronomical observation

Telescope pointing and celestial coordinates are often measured in arcseconds or even milliarcseconds. Use this tool to unify DMS into degrees or radians before doing trigonometric calculations to reduce cumulative errors from multiple manual conversions.

FAQ

How many radians is 1 degree? How many degrees is 1 radian?

1° = π/180 rad ≈ 0.0174533 rad; 1 rad = 180°/π ≈ 57.2958°. This is the fundamental conversion relationship between degrees and radians. All trigonometric function libraries (such as Python math, JavaScript Math) default to using radians as input.

Should I use degrees or radians for trigonometric functions?

The vast majority of programming languages, math libraries, and physics formulas use radians by default. For example, sin(π/2)=1 refers to radians; if you write sin(90), it is calculated as 90 radians, giving a completely different result. Degrees (°) are more commonly used only in engineering surveying, navigation, maps, and everyday communication.

How many degrees is 1 right angle? How many degrees is 1 full turn?

1 right angle = 90°, 1 turn = 360°. Therefore 1 turn = 4 right angles = 2π rad ≈ 6.2832 rad. Right angles are commonly used in geometric descriptions; turns (also called revolutions) are commonly used for rotation, motor RPM, and angle sensor readings.

How many arcminutes and arcseconds are in 1 degree?

1° = 60' (arcminutes), 1' = 60" (arcseconds), so 1° = 3600". For example, 30°15'20" converts to decimal degrees as 30 + 15/60 + 20/3600 ≈ 30.2556°. This degrees-minutes-seconds format is common in astronomical observation, geographic coordinates, and nautical positioning.

What is a milliradian (mrad) used for?

1 mrad = 0.001 rad ≈ 0.0573°. Milliradians are very convenient in ballistics, optical sighting, laser ranging, and small-angle approximation (sin x ≈ x), because 1 mrad at 100 m distance corresponds to approximately 10 cm of lateral offset, making estimation intuitive.

Is this converter free? Do I need to register?

Completely free, no registration or login required. Open the page and enter values directly; conversion results are displayed immediately, with no paywalls or account authorization requirements.

Will the data I enter be uploaded to a server?

No. All conversions are performed in the browser via JavaScript, with no network requests carrying your input values. Closing the tab immediately clears input data; nothing is recorded or analyzed.

How precise is the conversion?

All conversions use double-precision IEEE 754 floating-point numbers, with factors aligned to mathematical definitions (e.g., 1° = π/180 rad, 1 turn = 2π rad). The display layer retains 6 significant figures; the relative error for daily learning, engineering estimation, and programming debugging is far less than one part per million.

Can it be used offline?

Yes. After the page loads, all conversions run locally in the browser using built-in factors, with no API requests or network dependency. It works normally on airplanes, workshops, fieldwork, and other offline scenarios.

Why do degree/radian errors occur so easily in programming?

The common mistake is passing degrees directly to radian-based trigonometric functions. For example, in JavaScript, Math.sin(90) is not 1 — it computes sin(90 rad) ≈ 0.894. The correct approach is to first convert 90° to π/2 rad using this tool or code, then pass it to Math.sin.

How do I convert degrees-minutes-seconds in geographic coordinates to decimal degrees?

Divide minutes by 60, divide seconds by 3600, then add to degrees. For example, 39°54'27" = 39 + 54/60 + 27/3600 ≈ 39.9075°. This tool supports arbitrary conversion between degrees, arcminutes, and arcseconds; you can also first unify minutes/seconds into degrees, then convert to radians.

What's the difference between this page and the Unit Converter aggregator?

This page is a vertical deep-dive page for angle conversion, specifically covering 7 units across sexagesimal and radian systems, and includes angle knowledge, common pitfalls, and quick reference tables. The aggregator page is suitable for quickly switching between unrelated categories like length, mass, temperature; if you clearly only need angle conversion, this page is more direct.

Troubleshooting

Programming results are very different from expected?

The most common cause is passing degrees directly to radian-based trigonometric functions. Check your code: Math.sin(90) computes 90 rad; the correct approach is Math.sin(90 * Math.PI / 180) or Math.sin(Math.PI / 2).

After converting DMS to degrees, the value is clearly too large or too small?

Check whether minutes were divided by 60 and seconds by 3600. Incorrectly dividing seconds by 60 will make the result nearly 60 times too large; forgetting to add minutes/seconds to degrees will make the result too small.

Confusing milliradians with arcminutes causing errors?

1 mrad ≈ 3.4377' ≈ 0.0573°, 1' ≈ 0.2909 mrad. In shooting or optics, mrad is often used interchangeably with MOA (arcminute); remember that 1 mrad ≈ 3.44 MOA, not 1:1.

Results are displayed in scientific notation and hard to read?

When values are extremely small (e.g., arcseconds to radians), the page uses scientific notation to make numbers more readable. For example, 1" ≈ 4.848 × 10⁻⁶ rad; this is normal display formatting and does not represent precision loss.

Right angle/turn and degree conversions are wrong?

Confirm unit selection is correct: right angle = 90°, turn = 360°. Don't mistake "turn" for 1° or "right angle" for 1 turn; look carefully at the unit names when selecting from dropdowns.

Glossary

Degree (°)
The most commonly used angle unit. 1 turn = 360°, 1° = 60' = 3600". Widely used in engineering surveying, maps, navigation, and everyday communication.
Radian (rad)
SI standard angle unit. When arc length equals the radius, the subtended central angle is 1 rad. 1 turn = 2π rad ≈ 6.2832 rad, 1 rad ≈ 57.2958°. Default for math, physics, and programming.
Arcminute (')
Subdivision of the degree system, 1° = 60'. Commonly used in astronomical observation, geographic coordinates, and nautical positioning; for example, right ascension/declination are often expressed in DMS.
Arcsecond (")
Further subdivision of the degree system, 1' = 60", so 1° = 3600". Common in high-precision astronomical measurement and geodesy.
Milliradian (mrad)
One-thousandth of a radian, 1 mrad = 0.001 rad ≈ 0.0573°. Commonly used in shooting, optics, laser ranging, and small-angle approximation.
Right angle
Angle unit equal to 90°, i.e., 1/4 turn or π/2 rad. Most intuitive for describing perpendicular relationships in geometry.
Turn (revolution)
Angle unit equal to 360° or 2π rad, representing one complete rotation. Commonly used in motor RPM, encoder readings, and rotation animation.

Common Angle Conversion Anchors

FromToConversion Value
1°rad0.0174533 rad
1 rad°57.2958°
1'°0.0166667°
1"°0.0002778°
1 right angle°90°
1 turnrad6.2832 rad
1 mrad°0.0573°

Common Angle Values Reference Table

Degree (°)Radian (rad)Description
0°0 radZero angle
30°π/6 ≈ 0.5236 radSpecial angle, sin = 1/2
45°π/4 ≈ 0.7854 radSpecial angle, sin = cos = √2/2
60°π/3 ≈ 1.0472 radSpecial angle, cos = 1/2
90°π/2 ≈ 1.5708 radRight angle
180°π ≈ 3.1416 radStraight angle
270°3π/2 ≈ 4.7124 radReflex angle / three right angles
360°2π ≈ 6.2832 radFull turn

Privacy & Security

All angle conversions on this page run entirely in your browser. The values you enter never leave your device — no network requests, no server logs. Closing the page immediately clears all data.

Authoritative References