Angle Converter - Free Online Tool

Convert angle units instantly. Support for 5+ units including Degree, Radian, Gradian. Free calculator with accurate conversions, formula, and table.

📐 Angle Converter

Select your units from the lists below

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From Unit

Degree
Symbol: °

Radian
Symbol: rad

Gradian
Symbol: grad

Arcminute
Symbol: '

Arcsecond
Symbol: "

To Unit

Degree
Symbol: °

Radian
Symbol: rad

Gradian
Symbol: grad

Arcminute
Symbol: '

Arcsecond
Symbol: "

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📐 Angle Converter

Angle measurements appear in mathematics, engineering, and navigation. Convert between degrees, radians, gradians, and other angle units instantly. Whether you're working in trigonometry, engineering design, or navigation, our converter handles the calculations.

📐 About Angle

Angle measures the rotation or separation between two lines or surfaces meeting at a point. The most common units are degrees (360 per full circle), radians (2π per full circle), and gradians (400 per full circle). From mathematics and engineering to navigation and many scientific fields, angle measurements are used everywhere. Different fields use different units. Degrees remain widely used in everyday applications and navigation, while radians are preferred in mathematics and physics for their natural relationship with arc length. Gradians, though less common, offer decimal-based measurement. Working with all these units helps with scientific calculations, engineering design, and mathematical analysis.

Understanding Angle Measurement

Angle is a measure of rotation or the separation between two lines or surfaces that meet at a point. In the International System of Units (SI), the radian is the coherent unit for plane angle, defined as the angle subtended at the center of a circle by an arc equal in length to the radius. One full circle equals 2π radians, approximately 6.283 radians. The degree, equal to 1/360 of a full circle, is a non-SI unit accepted for use with the SI. The relationship between degrees and radians is fundamental: 360 degrees equals 2π radians, so one degree equals π/180 radians (approximately 0.017453 radians). The radian's mathematical elegance makes it the preferred unit in calculus and physics, as many trigonometric identities and derivatives are simplest when angles are expressed in radians.

📜 History & Context

The history of angle measurement dates back to ancient Babylonian astronomy around 2000 BCE. The Babylonians used a sexagesimal (base-60) number system and divided the circle into 360 parts, likely because 360 has many divisors and approximated the number of days in a year. This 360-degree system was adopted by Greek astronomers like Hipparchus in the 2nd century BCE, who further subdivided degrees into 60 arcminutes and arcminutes into 60 arcseconds. The radian, though conceptually older, was formally developed in the 18th century. The idea of measuring angles by the ratio of arc length to radius appears in the work of mathematicians like Roger Cotes (1714) and was implicit in Leonhard Euler's mathematical analysis. The term "radian" was coined by James Thomson in 1873 and adopted by Thomas Muir. The radian's mathematical advantages—particularly in calculus, where derivatives of trigonometric functions are simplest in radians—led to its adoption as the SI unit for plane angle in 1960, though degrees remain widely used in practical applications.

✨ Fun Facts

• The 360-degree circle comes from ancient Babylonian astronomy, likely chosen because 360 has many divisors and approximated the number of days in a year.
• One radian equals approximately 57.3 degrees—the angle where the arc length equals the radius of the circle.
• The term "radian" was coined in 1873 by James Thomson, though the concept had been used mathematically since the 18th century by Euler and others.
• The gradian, dividing a circle into 400 parts, was introduced during the French Revolution as part of decimalization efforts but saw limited adoption.
• In calculus, trigonometric function derivatives are simplest when angles are in radians—for example, the derivative of sin(x) is cos(x) only when x is in radians.

💡 Real-World Examples

• Engineering and construction: Building design, structural analysis, and surveying measure angles in degrees or radians. Converting between units helps with calculations and international standards.
• Mathematics and physics: Scientific calculations use radians for trigonometric functions and calculus. Converting to degrees helps with practical applications and reporting.
• Navigation and surveying: Compass bearings, map reading, and land surveying measure angles in degrees. Converting to radians helps with some mathematical calculations.
• Robotics and automation: Robot joint angles and mechanical positioning are measured in degrees or radians. Converting between units helps with control systems and programming.
• Astronomy and celestial navigation: Star positions and celestial coordinates use degrees, arcminutes, and arcseconds. Converting between units helps with calculations and reporting.
• Computer graphics: 3D modeling and animation measure rotation angles in degrees or radians. Converting between units helps with different software and programming languages.
• Mechanical engineering: Gear design, cam profiles, and linkage mechanisms measure angles in degrees or radians. Converting between units helps with design calculations.
• Geographic information systems: Map projections and coordinate systems use angles in various units. Converting between units helps with data processing and analysis.

🔧 Common Applications

• Mathematics and physics: Trigonometric calculations, calculus, and scientific analysis use radians. Converting to degrees helps with practical applications and reporting.
• Engineering and construction: Structural design, surveying, and mechanical systems measure angles in degrees or radians. Converting between units helps with calculations and standards.
• Navigation and mapping: Compass bearings, celestial navigation, and geographic coordinates use degrees. Converting to radians helps with some mathematical calculations.
• Computer science and graphics: Programming, 3D graphics, and game development use angles in different units. Converting between units helps with software compatibility and calculations.
• Robotics and automation: Joint angles, positioning systems, and control algorithms measure angles in various units. Converting between units helps with system integration and programming.

❓ Frequently Asked Questions

+ How do I convert degrees to radians?

To convert degrees to radians, multiply the degree value by π/180 (approximately 0.0174533). For example, 180 degrees equals π radians (approximately 3.14159), and 90 degrees equals π/2 radians (approximately 1.5708).

+ How many degrees are in a full circle?

There are exactly 360 degrees in a full circle. This is a standard measurement used in navigation, geometry, and everyday applications. For example, a right angle is 90 degrees, and a straight line is 180 degrees.

+ What is the relationship between degrees and radians?

A full circle contains 360 degrees or 2π radians. Therefore, 180 degrees equals π radians, and 90 degrees equals π/2 radians. Radians are commonly used in mathematics and physics, while degrees are more common in everyday applications.

+ How do I convert radians to degrees?

To convert radians to degrees, multiply the radian value by 180/π (approximately 57.2958). For example, π radians equals 180 degrees, and π/2 radians equals 90 degrees.

+ What is a radian?

A radian is the angle subtended at the center of a circle by an arc equal in length to the radius. One radian equals approximately 57.2958 degrees. There are 2π radians (approximately 6.28318) in a full circle. Radians are the preferred unit in mathematics and physics.