Angle Converter - Free Online Tool

Convert between degrees, radians, gradians, DMS, and 12 angle units instantly

Angles are one of the most fundamental measurements in mathematics, science, engineering, and everyday life. Whether you are a student working through trigonometry homework, a surveyor marking land boundaries, a photographer adjusting a camera lens, or an engineer designing mechanical parts, the ability to convert angles between different measurement systems is an essential skill. Our free Angle Converter makes that process instant and error-free. The most familiar unit of angle measurement is the degree (°). A full circle contains 360 degrees, a right angle is 90 degrees, and a straight line spans 180 degrees. This convention dates back to ancient Babylonian astronomy, which used a base-60 (sexagesimal) number system and divided the circle into 360 equal parts to roughly match the number of days in a year. Degrees remain the dominant unit in everyday contexts, navigation, and most engineering disciplines. The radian (rad) is the standard unit of angle measurement in mathematics and physics. One radian is defined as the angle subtended at the center of a circle by an arc whose length equals the radius of that circle. Because a full circle has circumference 2π × r, a complete revolution equals 2π radians ≈ 6.2832 radians. The conversion relationship — 1 radian = 180/π ≈ 57.2958 degrees — appears constantly in calculus, signal processing, and physics equations where trigonometric functions behave most naturally with radian input. The gradian (grad or gon) divides the circle into 400 equal parts, making a right angle exactly 100 gradians. This neat decimal relationship was designed during the French Revolution as part of the metric system movement. Gradians are still widely used in surveying, particularly in continental Europe, because the 400-part system makes it easy to work with slopes and coordinate geometry. For precision angle work — such as astronomy, navigation, and geodesy — degrees are subdivided into arcminutes (1/60 of a degree) and arcseconds (1/3600 of a degree). The DMS (Degrees, Minutes, Seconds) format is used on maps, GPS coordinates, and telescope pointing systems. Our converter supports direct DMS input, automatically converting your compound angle to decimal degrees and then into any other unit. Specialized angle units include the NATO mil (1/6400 of a circle, used in military artillery calculations), the milliradian (1/1000 of a radian, used in rifle scope adjustments and ballistics), the revolution (a full 360-degree turn), the quadrant (90 degrees), the sextant (60 degrees, the angle measured by the navigation instrument of the same name), and the sign (30 degrees, used in astrology and traditional astronomy to divide the zodiac into 12 equal parts). Our Angle Converter provides all these units in one place. Enter a value in decimal form or switch to DMS input mode, select your source unit, and instantly see the result in your target unit along with a complete all-units table so you can compare every equivalent simultaneously. A visual angle diagram shows how the angle looks on a circle, and a step-by-step formula breakdown shows exactly how the conversion was computed. Use the preset buttons for common angles like 30°, 45°, 90°, or 180°, and the swap button to quickly reverse the conversion direction.

Understanding Angle Units

What Is an Angle?

An angle is the measure of rotation between two rays that share a common endpoint called the vertex. Angles describe the amount of turn needed to move from one ray's direction to the other. They appear in geometry (polygon interior angles), trigonometry (sine, cosine, tangent functions), physics (rotational motion, wave cycles), navigation (compass bearings, latitude and longitude), and engineering (gear ratios, structural slopes). Angles can be measured in many unit systems — degrees, radians, gradians, and more — each suited to particular applications. A full rotation is 360 degrees, 2π radians, or 400 gradians. Understanding the relationships between these systems allows you to work fluently across mathematics, science, and practical fields.

How Are Angles Converted?

All angle unit conversions use degrees as an intermediate base unit. Each unit has a fixed number of degrees equivalent: 1 radian = 180/π ≈ 57.2958°, 1 gradian = 0.9°, 1 arcminute = 1/60°, 1 arcsecond = 1/3600°, 1 revolution = 360°, 1 quadrant = 90°, 1 sextant = 60°, 1 sign = 30°, 1 NATO mil = 360/6400 ≈ 0.05625°, 1 milliradian = 180/(1000π) ≈ 0.05730°. To convert from unit A to unit B: multiply the input by A's degrees-per-unit factor to get degrees, then divide by B's degrees-per-unit factor. For DMS input, first convert to decimal degrees using DD = D + M/60 + S/3600, then apply the standard conversion. The result can also be expressed back in DMS form using the reverse formula.

Why Does Unit Choice Matter?

Choosing the right angle unit can simplify calculations significantly. In calculus, derivatives of trigonometric functions like sin and cos are cleanest when angles are in radians — d/dx sin(x) = cos(x) only holds in radians, not degrees. In surveying, gradians simplify slope calculations because a 1% slope equals a 1-gradian angle difference from horizontal. In military applications, mils allow easy mental arithmetic for range estimation: an object 1 meter wide at 1000 meters distance subtends approximately 1 milliradian. In GPS and astronomy, arcminutes and arcseconds express latitude and longitude with fine precision. Understanding which unit fits your context saves conversion steps and reduces the chance of error in calculations.

Precision and Practical Notes

This converter handles angles of any magnitude, including values greater than 360 degrees (full rotations) or negative angles (clockwise direction). The visual angle diagram normalizes the input to a 0–360 degree range for display purposes. Conversion results are shown with up to 8 significant figures, which exceeds the practical precision needed for most applications. For very small angles (below 10⁻⁶ degrees), values are displayed in scientific notation. Note that the NATO mil (1/6400 circle) differs from the Soviet/Warsaw Pact mil (1/6000 circle) and the Swedish streck (1/6300 circle) — this converter uses the NATO standard. For DMS input, minutes and seconds should each be in the range 0–59.

Angle Conversion Formulas

Degrees to Radians

radians = degrees × π / 180

Multiply the angle in degrees by π/180 (approximately 0.017453) to convert to radians.

Radians to Degrees

degrees = radians × 180 / π

Multiply the angle in radians by 180/π (approximately 57.2958) to convert to degrees.

Degrees to Gradians

gradians = degrees × 10 / 9

Multiply degrees by 10/9 (approximately 1.1111) to convert to gradians. A right angle is 100 gradians.

Degrees to Turns (Revolutions)

turns = degrees / 360

Divide the angle in degrees by 360 to express it as a fraction of a full revolution.

Angle Conversion Reference Tables

Common Angle Conversions

Standard angles from 0° to 360° with their equivalents in radians, gradians, and turns.

Degrees (°)	Radians (rad)	Gradians (gon)	Turns (rev)
0	0	0	0
15	π/12 ≈ 0.2618	16.667	0.04167
30	π/6 ≈ 0.5236	33.333	0.08333
45	π/4 ≈ 0.7854	50	0.125
60	π/3 ≈ 1.0472	66.667	0.16667
90	π/2 ≈ 1.5708	100	0.25
120	2π/3 ≈ 2.0944	133.333	0.33333
180	π ≈ 3.1416	200	0.5
270	3π/2 ≈ 4.7124	300	0.75
360	2π ≈ 6.2832	400	1

Worked Examples

Convert 45° to Radians

A trigonometry problem requires the angle 45° expressed in radians for use in a calculus formula.

Use the formula: radians = degrees × π / 180

Substitute: radians = 45 × π / 180

Simplify: radians = π / 4 ≈ 0.7854

45° equals π/4 radians (approximately 0.7854 rad). This is one of the most commonly used angles in trigonometry, where sin(π/4) = cos(π/4) = √2/2.

Convert π/3 Radians to Degrees

A physics textbook gives a launch angle of π/3 radians. You need the equivalent in degrees for a presentation.

Use the formula: degrees = radians × 180 / π

Substitute: degrees = (π/3) × 180 / π

Simplify: degrees = 180 / 3 = 60

π/3 radians equals exactly 60°. This is one of the special angles where sin(60°) = √3/2 and cos(60°) = 1/2.

Convert 90° to Gradians and NATO Mils

A surveyor needs to express a right angle in gradians for European equipment and NATO mils for a military report.

Gradians: 90 × 10/9 = 100 gradians

NATO mils: 90 / 360 × 6400 = 1600 mils

Verify: 100 gradians = 100/400 × 360° = 90° ✓

90° equals exactly 100 gradians (a convenient round number by design) and 1,600 NATO mils.

How to Use the Angle Converter

Choose Your Input Mode

Select Decimal mode to enter a single number (e.g., 45.5 degrees), or switch to DMS mode to enter an angle as Degrees, Minutes, and Seconds (e.g., 45° 30' 0"). DMS mode always converts from degrees to your chosen target unit.

Set From and To Units

Use the From Unit and To Unit dropdowns to select your source and target angle units. Available units include degrees, radians, gradians, arcminutes, arcseconds, revolutions, quadrants, sextants, signs, octants, NATO mils, and milliradians. Use the swap arrow button to instantly reverse the direction.

Enter a Value or Use a Preset

Type your angle value in the input field, or click one of the Common Angle Presets (0°, 30°, 45°, 60°, 90°, 120°, 180°, 270°, 360°) for frequently used angles. The conversion updates automatically as you type — no need to press Convert.

Read Results and Export

The main result shows your converted value with the formula used. Below that, a visual diagram shows the angle on a circle, and the All Unit Equivalents table lists the angle in every supported unit. Use the copy icon on any row to copy that value, or click Export CSV to download all results as a spreadsheet.

Frequently Asked Questions

How do I convert degrees to radians?

To convert degrees to radians, multiply the degree value by π/180 (approximately 0.017453). For example, 90° × π/180 = π/2 ≈ 1.5708 radians. Conversely, to convert radians to degrees, multiply by 180/π (approximately 57.2958). So 1 radian ≈ 57.2958°. These relationships arise because a full circle equals 360° and also 2π radians — dividing both by 360 gives 1° = π/180 radians. Our converter applies this formula automatically, showing the exact multiplication factor used so you can verify the calculation yourself.

What is a gradian (gon), and when is it used?

A gradian (also called a gon or grad) divides a full circle into 400 equal parts, so a right angle equals exactly 100 gradians. This unit was introduced during the French Revolution as part of an effort to decimalize measurement systems. Gradians are still widely used in land surveying and civil engineering, particularly in continental Europe. Because 400 is a round number that aligns with percentage-based slope calculations, surveyors find the gradian system convenient for computing horizontal and vertical offsets. The conversion is straightforward: 1 gradian = 0.9 degrees, and 1 degree = 10/9 gradians ≈ 1.1111 gradians.

What is the difference between arcminutes and arcseconds?

Arcminutes and arcseconds are subdivisions of degrees used for precise angular measurements. One arcminute (') equals 1/60 of a degree, and one arcsecond (") equals 1/3600 of a degree (or 1/60 of an arcminute). These units are used in astronomy to describe the apparent size of celestial objects (the full Moon is about 30 arcminutes wide), in navigation for GPS coordinates (1 arcminute of latitude ≈ 1 nautical mile), and in optics for describing angular resolution. The DMS (Degrees, Minutes, Seconds) format, like 40° 26' 47", is the standard way to express latitude and longitude on maps and GPS devices.

What is a NATO mil and how does it differ from a milliradian?

The NATO mil divides a full circle into exactly 6,400 parts, so 1 NATO mil = 360/6400 = 0.05625 degrees. The milliradian (mrad) is 1/1000 of a radian, which equals approximately 0.05730 degrees — slightly larger than a NATO mil. The mil system was designed so that at a range of 1,000 meters, 1 mil corresponds to approximately 1 meter of lateral movement, making range and windage calculations easier for artillery and rifle scopes. Note that the Soviet mil uses 6,000 divisions per circle and the Swedish streck uses 6,300 — all three give slightly different values for 'one mil.' This converter uses the NATO standard of 6,400 mils per circle.

What does the DMS input mode do?

DMS stands for Degrees, Minutes, Seconds — a compound angle format used in navigation, cartography, astronomy, and surveying. Instead of writing 40.4464°, you can express the same angle as 40° 26' 47.04". In DMS mode, our converter accepts three separate fields for degrees, minutes, and seconds, then combines them into a decimal degree value using the formula: decimal degrees = D + M/60 + S/3600. The result is then converted to your chosen target unit. DMS mode always treats the input as degrees before converting, so the From Unit selector is disabled in that mode.

Why does the visual diagram only show 0–360°?

The circular angle diagram normalizes any input angle to the 0–360 degree range for display purposes, because a circle can only show one full revolution visually. If you enter 450 degrees, the diagram shows the equivalent position at 90 degrees (450 mod 360 = 90). Similarly, negative angles are mapped to their positive circular equivalent (for example, −90° maps to 270°). This normalization only affects the visual; the actual converted result shown in the main output and the all-units table uses the full, unnormalized value. For angles greater than 360 degrees, the result accurately reflects the multiple-revolution value (for example, 720 degrees = 4π radians, not 2π).