Angle Converter

Complete Guide to Angle Conversion

Angle conversion is essential across mathematics, engineering, physics, navigation, astronomy, surveying, construction, and countless applications where angular measurements are essential. Whether you're calculating trigonometric functions in radians, converting degrees to radians for calculus, converting between different angle units, working with geographic coordinates in degrees minutes and seconds, converting between degrees and gradians for surveying, converting between different angle units, or converting between different angle units, our free online angle converter makes it easy to convert between Degree, Radian, Grad, Gon, Minute, Second, Sign, Mil, Revolution, Circle, Turn, Quadrant, Right angle, Sextant, and other angle units instantly. Understanding angle units and conversions is crucial for accurate measurements in mathematics, engineering, physics, navigation, astronomy, and international communication.

Understanding Angle Measurement Systems

Angles measure rotation or the space between two intersecting lines. The most common angle units are degrees, radians, gradians (also called gons), arcminutes, and arcseconds. Each system has its historical roots and specific applications. The degree system, dating back to ancient Babylonian mathematics, divides a circle into 360 equal parts, making it intuitive for everyday use. The radian system, based on the radius of a circle, is the standard unit in mathematics and physics because it simplifies trigonometric calculations. The gradian system, which divides a circle into 400 parts, is used in surveying and some European engineering applications.

Standard Angle Units

Degree (°) - The Most Common Unit

A Degree is the most familiar unit of angle measurement. One complete revolution equals 360 degrees, likely based on the approximate number of days in a year. Degrees are subdivided into arcminutes (') and arcseconds ("), where 1 degree = 60 arcminutes = 3,600 arcseconds. This sexagesimal system (base-60) has been used since ancient times. Degrees are ideal for navigation, geography (latitude and longitude are measured in degrees), construction, and general angular measurements.

Radian (rad) - The SI Derived Unit

A Radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius. One complete circle equals 2π radians, approximately 6.28318 radians. The radian is the SI (International System of Units) derived unit for angle measurement. Radians are preferred in mathematics, physics, and engineering because they make calculus and trigonometry formulas simpler—the derivative of sin(x) is cos(x) only when x is in radians. One radian equals approximately 57.2958 degrees.

Grad and Gon - Metric Angle Units

A Grad (also called Gon) divides a circle into 400 equal parts, making it a metric-friendly angle unit. One complete revolution equals 400 gradians, so a right angle equals 100 gradians. Gradians are commonly used in surveying, civil engineering, and some European countries for mapping and construction. The gradian system was developed as part of the metric system movement, providing a decimal-based alternative to the 360-degree system. One gradian equals 0.9 degrees.

Minute (') and Second (") - Arc Subdivisions

Arcminutes (') and Arcseconds (") are subdivisions of degrees used for precise angular measurements, especially in astronomy, navigation, and geodesy. One degree contains 60 arcminutes, and one arcminute contains 60 arcseconds. This creates a hierarchical system: 1° = 60' = 3,600". Arcminutes and arcseconds are essential for celestial navigation, where star positions must be measured with extreme precision.

Specialized Angle Units

Sign

A Sign equals 30 degrees, representing one-twelfth of a circle. This unit is historically used in astrology and astronomy, where the zodiac is divided into 12 signs, each representing 30 degrees of the celestial sphere.

Mil (Angular Mil)

A Mil (angular mil) equals 0.05625 degrees, where 6,400 mils equal 360 degrees (one full circle). This unit is commonly used in military applications for artillery and navigation, providing a precise way to measure angles in mils rather than degrees.

Revolution, Circle, and Turn

Revolution, Circle, and Turn are all equivalent units representing one complete rotation, equal to 360 degrees or 2π radians. These units are used in rotational mechanics, engineering, and when describing complete rotations or cycles.

Quadrant and Right Angle

Quadrant and Right Angle are equivalent units representing one-quarter of a circle, equal to 90 degrees or π/2 radians. These units are fundamental in geometry and trigonometry, representing perpendicular angles.

Sextant

A Sextant equals 60 degrees, representing one-sixth of a circle. Historically, sextants were navigation instruments used to measure angles, and this unit reflects that measurement system.

How to Convert Degree to Radian

Converting Degree to Radian is one of the most common angle conversions, especially in mathematics, physics, and engineering. To convert from Degree to Radian, multiply the degree value by π/180 (approximately 0.0174533). This conversion factor comes from the definition: 1 radian = 180/π degrees ≈ 57.2958 degrees, so 1 degree = π/180 radians. For example, 90° × (π/180) = π/2 radians ≈ 1.5708 radians. This conversion is essential for understanding angular measurements, where degrees are used in everyday applications while radians are used in mathematical calculations, especially in calculus and trigonometry. The relationship is straightforward: simply multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. For quick mental calculations, you can approximate 1 degree ≈ 0.0175 radians, though our converter uses the exact conversion factor (π/180) for precision. This conversion is particularly important when working with trigonometric functions, calculating derivatives and integrals, understanding rotational motion, analyzing waveforms, and performing mathematical calculations that require radians.

Angle vs Rotation: Understanding the Relationship

One of the most important concepts in angle conversion is understanding the relationship between angle and rotation:

• Angle: A measure of the space between two intersecting lines or the amount of rotation, measured in degrees, radians, or other angle units (e.g., 90°, π/2 rad).
• Rotation: The actual turning or spinning motion, often measured in complete revolutions or turns (e.g., 1 revolution = 360° = 2π rad).

When converting between angle units, we're working with the magnitude of angular measurement, not the physical rotation itself. For example, converting "90 degrees" to "π/2 radians" represents the same angle, just expressed in different units. Understanding this distinction helps clarify that angle conversions measure angular space, while rotation involves the actual motion or turning.

Angle Conversion Formulas

Our angle converter uses these precise formulas for accurate conversions:

Angle Conversion Table

Quick reference for converting between common angle units:

Common Angle Conversion Examples

Frequently Asked Questions (FAQ)

Real-World Applications

Mathematics and Trigonometry

Angle conversion is fundamental in mathematics for trigonometric functions, calculus, and geometry. Converting between degrees and radians is essential when working with trigonometric functions (sin, cos, tan), as most mathematical functions require radians. Understanding angle conversions helps students and professionals solve problems involving angles, rotations, and circular motion.

Engineering and Construction

Engineering and construction require angle conversions for CAD drawings, structural design, and surveying. Converting between degrees (used in specifications and drawings) and radians (used in calculations) helps engineers design structures, analyze forces, and ensure accurate measurements. Surveying applications often use gradians for precise angle measurements.

Navigation and Geography

Navigation and geography require angle conversions for latitude and longitude coordinates, compass directions, and celestial navigation. Converting between degrees, arcminutes, and arcseconds helps navigators determine positions, plot courses, and understand geographic coordinates. GPS systems use degrees, minutes, and seconds for precise location specifications.

Astronomy and Physics

Astronomy and physics require angle conversions for celestial positions, rotational motion, and angular measurements. Converting between degrees, arcminutes, and arcseconds helps astronomers measure star positions, planet movements, and celestial object sizes. Physics applications use radians for angular velocity, angular acceleration, and rotational dynamics.

Related Unit Converters

Explore our other unit conversion tools for comprehensive measurement solutions:

• Angular Velocity Converter - Convert radians per second, degrees per second angular velocity
• Angular Acceleration Converter - Convert radians per second squared angular acceleration
• Length Converter - Convert meters, feet, inches length units
• Area Converter - Convert square meters, square feet area units
• Volume Converter - Convert liters, gallons volume units
• Speed Converter - Convert km/h, mph, m/s speed units
• Frequency & Wavelength Converter - Convert frequency and wavelength units
• Time Converter - Convert seconds, minutes, hours time units
• Weight & Mass Converter - Convert kilograms, pounds weight units
• Temperature Converter - Convert Celsius, Fahrenheit temperature units
• Force Converter - Convert newtons, pounds-force force units
• Energy Converter - Convert Joules, Calories energy units

Tips for Angle Conversion

• Quick Approximation: For degrees to radians, multiply by 0.0175 (exact: π/180 ≈ 0.0174533)
• Remember Key Values: 180° = π rad, 90° = π/2 rad, 360° = 2π rad, 1° = 60 arcmin = 3,600 arcsec
• Common Conversions: 30° = π/6 rad, 45° = π/4 rad, 60° = π/3 rad, 90° = π/2 rad
• Gradian System: 1 grad = 0.9°, 90° = 100 grad, 360° = 400 grad
• Arc Subdivisions: 1° = 60' = 3,600", 1' = 60"
• Circle Fractions: 1 quadrant = 90°, 1 sextant = 60°, 1 sign = 30°
• Military Units: 1 mil = 0.05625°, 6,400 mils = 360°