Angle Converter

About Angle Converter

Welcome to the Angle Converter, a precision instrument for converting between all major angle units. This interactive tool features a visual protractor diagram that animates in real-time as you type, instant bidirectional conversion across 7 different units, and DMS (Degrees-Minutes-Seconds) support. Whether you are a student, engineer, surveyor, astronomer, or developer, this converter provides accurate, professional-grade angle conversions.

Supported Angle Units

This converter supports the following angle measurement units:

Conversion Formulas

Degrees to Radians

Degrees to Gradians

Degrees to Turns

Degrees to Arcminutes and Arcseconds

Degrees to Milliradians

Understanding DMS Notation

DMS (Degrees-Minutes-Seconds) is a notation for expressing angles using degrees (°), arcminutes (′), and arcseconds (″). This system is analogous to hours, minutes, and seconds for time.

• 1 degree (°) = 60 arcminutes (′)
• 1 arcminute (′) = 60 arcseconds (″)
• 1 degree (°) = 3,600 arcseconds (″)

For example, 45.5° in DMS notation is 45° 30′ 0″ (45 degrees, 30 minutes, 0 seconds).

How to Use This Converter

1. Quick presets: Click any preset button (30°, 45°, 90°, etc.) to instantly see conversions for common angles.
2. Enter any value: Type a number in any unit field. All other fields update instantly with converted values.
3. Watch the visual: The protractor diagram animates to show your angle, helping you visualize its size.
4. Use DMS input: For precise angular measurements, enter degrees, minutes, and seconds separately.
5. Check the fraction: See what fraction of a full circle your angle represents.

Common Angle References

Applications of Angle Units

Degrees (°)

The most widely used angle unit, degrees are familiar and intuitive. A full circle is divided into 360 equal parts. This Babylonian-derived system remains the standard in education, construction, navigation, and everyday use.

Radians (rad)

The SI unit for angles, radians are essential in mathematics and physics. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius. Key relationships include:

• Circular motion: arc length s = rθ (where θ is in radians)
• Calculus: derivatives of sin(x) and cos(x) are simplest when x is in radians
• Physics: angular velocity ω is measured in rad/s

Gradians (gon)

Also called gons or grades, gradians divide a right angle into 100 parts and a full circle into 400 parts. This decimal-based system simplifies calculations in surveying and civil engineering, as percentage grades and slopes align naturally with gradians.

Turns

A turn (also called a revolution, cycle, or full rotation) represents one complete 360° rotation. Turns are intuitive for describing rotational motion, gear ratios, and cyclic phenomena. In computer graphics and game development, rotations are often expressed in turns or fractions of turns.

Arcminutes and Arcseconds

These subdivisions of degrees provide precision for astronomical and geographic measurements:

• Astronomy: Star positions, planetary motions, and telescope specifications use arcminutes and arcseconds
• Navigation: One nautical mile equals one arcminute of latitude
• Cartography: Map coordinates often use DMS notation

Milliradians (mrad)

Milliradians are widely used in military and ballistics applications because of a convenient property: at 1000 meters distance, 1 milliradian subtends approximately 1 meter. This makes range estimation and trajectory calculations intuitive without complex mathematics.

Frequently Asked Questions

How do I convert degrees to radians?

To convert degrees to radians, multiply the degree value by π/180. For example, 90° × (π/180) = π/2 radians ≈ 1.5708 radians. This calculator performs this conversion instantly as you type.

What is the difference between degrees and gradians?

Degrees divide a full circle into 360 parts, while gradians (also called gons or grades) divide it into 400 parts. Gradians are commonly used in surveying because 100 gradians equals a right angle, making calculations easier. 1 gradian = 0.9 degrees.

What is a turn in angle measurement?

A turn (also called a revolution or full rotation) represents one complete 360° rotation. So 1 turn = 360° = 2π radians = 400 gradians. Half a turn is 180°, and a quarter turn is 90°.

How are arcminutes and arcseconds related to degrees?

Arcminutes and arcseconds are subdivisions of degrees. 1 degree = 60 arcminutes (′), and 1 arcminute = 60 arcseconds (″). Therefore, 1 degree = 3600 arcseconds. These units are commonly used in astronomy, navigation, and cartography for precise angle measurements.

What are milliradians used for?

Milliradians (mrad or mil) are used primarily in military applications, ballistics, and optics. 1 milliradian ≈ 0.0573°. At 1000 meters distance, 1 milliradian subtends approximately 1 meter, making range estimation straightforward.

Additional Resources

• Angle - Wikipedia
• Radian - Wikipedia
• Gradian - Wikipedia
• Arcminute and Arcsecond - Wikipedia
• Milliradian - Wikipedia

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