Angle Converter
Convert between degrees, radians, gradians, turns, arcminutes, arcseconds, and milliradians instantly. See all results at once with a visual arc display and reference tables.
Visual Angle Display
See the current angle rendered as an arc. The shaded region represents the angle magnitude (displayed modulo 360°).
Common Angle Reference Table
| Description | Degrees | Radians | Gradians | Turns |
|---|---|---|---|---|
| Zero angle | 0° | 0 | 0 grad | 0 |
| Common trig (30°) | 30° | π/6 ≈ 0.5236 | 33.333 grad | 1/12 |
| Common trig (45°) | 45° | π/4 ≈ 0.7854 | 50 grad | 1/8 |
| Common trig (60°) | 60° | π/3 ≈ 1.0472 | 66.667 grad | 1/6 |
| Right angle | 90° | π/2 ≈ 1.5708 | 100 grad | 1/4 |
| Straight angle | 180° | π ≈ 3.1416 | 200 grad | 1/2 |
| Three-quarter turn | 270° | 3π/2 ≈ 4.7124 | 300 grad | 3/4 |
| Full circle | 360° | 2π ≈ 6.2832 | 400 grad | 1 |
Key Features
Seven Units at Once
See your angle converted to degrees, radians, gradians, turns, arcminutes, arcseconds, and milliradians simultaneously. No need for multiple conversions.
Visual Arc Display
An interactive SVG arc renders the current angle in real time so you can visually understand the magnitude of the value you entered.
Reference Table
Built-in table covering right angles, straight angles, full circles, and common trigonometric angles like 30, 45, and 60 degrees for quick lookup.
Instant Results
Conversions update as you type, providing real-time feedback without page reloads or button clicks. Swap units with a single tap.
How Angle Converter Works
- Enter the angle value you want to convert in the input field.
- Select the source unit (degrees, radians, gradians, turns, arcminutes, arcseconds, or milliradians) from the "From" dropdown.
- All seven unit conversions appear simultaneously in the result area below.
- Use the swap button (↔) to quickly reverse the from and to units.
- Check the visual arc display to see a graphical representation of your angle.
- Click "Copy" to copy all conversion results to your clipboard.
Radians are the ratio of arc length to radius, which is why they appear naturally in calculus. When you take the derivative of sin(x), the result is cos(x) only when x is in radians. Using degrees introduces an extra factor of pi/180 that complicates every formula in physics and engineering.
JavaScript's Math.sin(), Math.cos(), and Math.atan() functions all use radians, not degrees. Passing 90 to Math.sin() gives 0.894 instead of the expected 1. Always convert degrees to radians first with (degrees * Math.PI / 180) before using trigonometric functions in code.
Understanding Angle Measurement
Angles are fundamental geometric quantities that describe the rotation between two rays sharing a common endpoint, called the vertex. From the earliest days of astronomy and navigation to modern engineering, computer graphics, and robotics, accurate angle measurement is essential. Different fields have adopted different units for measuring angles, each optimized for particular applications. This converter supports the seven most widely used angle units, letting you move between them instantly.
Degrees are the most familiar angle unit in everyday life. The system divides a full circle into 360 equal parts, a convention that dates back to ancient Babylonian astronomers who used a base-60 number system. The choice of 360 is particularly convenient because it has many divisors (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360), making it easy to subdivide a circle into equal parts. Degrees remain the standard unit for navigation, architecture, everyday geometry, and most educational settings around the world.
Radians are the preferred unit in mathematics and the physical sciences. One radian is the angle subtended at the center of a circle by an arc whose length equals the circle's radius. Since the circumference of a circle is two times pi times the radius, a full circle contains exactly 2π radians, approximately 6.2832. Radians make many mathematical formulas simpler and more elegant. In calculus, the derivative of the sine function is the cosine function only when the argument is measured in radians. This natural connection to the circle's geometry is why radians are the SI unit for plane angles and the default in most programming languages and scientific calculators.
Gradians, also known as gons or grades, divide a full circle into 400 parts. This means a right angle equals exactly 100 gradians, which aligns neatly with the decimal system. Gradians were introduced during the French Revolution as part of the broader push to decimalize measurement. They are still used today in surveying, land measurement, and some European engineering applications. Surveying instruments often include a gradian mode alongside degrees because the round numbers simplify fieldwork calculations involving perpendicular and parallel lines.
Turns represent the most intuitive angle unit: one turn equals one complete rotation, or 360 degrees. While less common in formal scientific work, turns appear in engineering contexts, frequency analysis (cycles per second), and everyday language ("half turn," "quarter turn"). Some programming frameworks and game engines use turns for rotation because the values stay between 0 and 1 for a single revolution, making interpolation straightforward.
Minutes and seconds of arc provide finer subdivisions of degrees, following the same 60-based system used for time. One degree contains 60 arcminutes, and one arcminute contains 60 arcseconds. This DMS (degrees, minutes, seconds) notation is the standard in navigation, astronomy, and cartography. Geographic coordinates (latitude and longitude) are traditionally expressed in DMS format. In astronomy, arcseconds measure the angular diameter of celestial objects and the resolving power of telescopes.
Milliradians are one-thousandth of a radian and are widely used in military, ballistics, and precision optics applications. The practical appeal of milliradians comes from a useful approximation: at a distance of 1000 meters, one milliradian subtends approximately one meter. This makes range estimation and scope adjustment calculations intuitive. NATO and most modern military forces use milliradians for artillery fire control, and rifle scopes frequently feature milliradian-based adjustment turrets for long-range shooting.
When to Use This
Game Developer
A game programmer rotates sprites using a physics engine that expects radians, but the level editor exports rotation data in degrees. Converting accurately prevents objects from spinning to wrong orientations and ensures collision detection remains aligned.
Land Surveyor
A surveyor measures bearing angles in degrees, minutes, and seconds (DMS) from a total station but must enter decimal degrees into GIS software. Precise conversion ensures property boundaries are plotted accurately down to the centimeter level.
Mechanical Engineer
A CNC machinist programs tool paths using gradians (common in European machining standards) but the CAD software outputs angles in degrees. Converting between these units ensures the cutting tool follows the exact geometry specified in the blueprint.
FAQ
Why do mathematicians prefer radians over degrees?
Radians are the natural unit for angles in mathematics because they emerge directly from the geometry of a circle. When angles are measured in radians, key calculus results hold in their simplest form. For instance, the derivative of sin(x) is cos(x) without any additional conversion factor. The Taylor series expansions for trigonometric functions also assume radian input. Beyond calculus, radians simplify the relationship between arc length (s), radius (r), and angle (θ): s = rθ. This elegance is why radians are the SI derived unit for angles and the default in virtually every scientific computing environment.
What is the difference between degrees and gradians?
Both degrees and gradians divide a full circle into equal parts, but they differ in how many parts. Degrees split the circle into 360 parts, while gradians divide it into 400 parts. This means a right angle is 90 degrees but 100 gradians. Gradians were designed to align with the decimal system, making mental arithmetic easier in certain contexts. They are primarily used in surveying, civil engineering, and some European countries. For most other purposes, including education, navigation, and general science, degrees remain the dominant choice.
How do I read a protractor to measure angles?
To measure an angle with a protractor, place the center mark of the protractor precisely on the angle's vertex (the point where the two rays meet). Align one ray with the 0-degree baseline along the bottom edge of the protractor. Then read the value where the second ray crosses the protractor's curved scale. Most protractors have two sets of numbers (an inner scale and an outer scale) running in opposite directions. Use the scale that starts from 0 on the ray you aligned with the baseline. If the angle opens to the left, use the outer scale, and if it opens to the right, use the inner scale.
What is DMS format for angles?
DMS stands for Degrees, Minutes, Seconds, which is a way to express angles using subdivisions of a degree. One degree equals 60 minutes of arc, and one minute of arc equals 60 seconds of arc. For example, an angle of 45.7625 degrees converts to 45° 45′ 45″ in DMS. This format is standard in navigation (GPS coordinates), cartography, and astronomy because it provides precision without resorting to long decimal strings. To convert from DMS to decimal degrees, add the degrees to the minutes divided by 60 and the seconds divided by 3600.
What are the full circle values in each angle unit?
A complete rotation around a circle has different numerical values depending on the unit used. In degrees it is 360, in radians it is 2π (approximately 6.2832), in gradians it is 400, in turns it is 1, in arcminutes it is 21,600, in arcseconds it is 1,296,000, and in milliradians it is approximately 6,283.19. Knowing these full-circle values is useful for quick mental conversions and for verifying that your converter results are correct.
What are milliradians and where are they used?
A milliradian (mrad) is one-thousandth of a radian, equal to approximately 0.0573 degrees. Milliradians are heavily used in military and tactical applications because of a practical property: at a distance of 1,000 units (meters, yards, etc.), one milliradian subtends approximately one unit of length. This makes it easy to estimate distances and adjust aim without complex calculations. Rifle scopes, artillery fire-control systems, and NATO standard targeting all use milliradians. They are also found in some precision engineering and optics applications where small angular adjustments need to be specified clearly.
How do I convert between degrees and radians quickly?
The key relationship is that 180 degrees equals π radians. To convert degrees to radians, multiply by π/180. To go from radians to degrees, multiply by 180/π. For mental math, memorize a few anchor points: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, and 360° = 2π. You can also use the rough approximation that 1 radian is about 57.3 degrees, so multiplying a radian value by 57.3 gives a quick degree estimate.
Angle Units in Different Fields
Angles are measured differently depending on the discipline. While degrees are the most familiar unit in everyday life, mathematicians, engineers, surveyors, and astronomers each have their own preferred systems optimized for their specific calculations. Understanding when and why each unit is used helps you choose the right one for your work and avoid costly conversion errors.
| Field | Degrees (°) | Radians (rad) | Gradians (gon) | Turns |
|---|---|---|---|---|
| Mathematics | ○ | ★ Primary | ○ | ○ |
| Physics | ○ | ★ Primary | — | ○ |
| Civil Engineering | ○ | — | ★ Primary | — |
| Surveying | ○ | — | ★ Primary | — |
| Navigation | ★ Primary | — | — | — |
| Astronomy | ★ Primary | ○ | — | — |
| Game Development | ★ Primary | ○ | — | ○ |
★ = standard unit in the field • ○ = commonly used • — = rarely used
Degrees divide a full rotation into 360 equal parts, a convention dating back to ancient Babylonian astronomy. The number 360 was chosen because it is highly divisible (by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180), making fractional angles easy to work with. Degrees remain the default in navigation, astronomy, architecture, and everyday conversation.
Radians define angle as the ratio of arc length to radius, making them dimensionless and mathematically natural. A full circle equals 2π radians. All calculus-based trigonometric formulas assume radian input, which is why every programming language's math library (sin, cos, tan) uses radians by default. Attempting to pass degrees into these functions is one of the most common bugs in scientific computing.
Gradians (also called gons) divide a right angle into exactly 100 parts, making a full circle 400 gradians. This decimal-friendly system is standard in European surveying and civil engineering because slope calculations become simple percentages. Most scientific calculators include a gradian mode alongside degree and radian modes.
Use the angle converter above to translate between all these units instantly and avoid the common pitfalls of manual conversion, especially the tricky degrees-to-radians calculation that trips up even experienced programmers.