Angle Converter

An angle is a geometric figure formed by two rays (or lines) sharing a common endpoint, called the vertex. Angles measure the amount of rotation between these rays and are fundamental to geometry, trigonometry, physics, engineering, and astronomy.

The degree (symbol °) is the most commonly used unit of angular measurement. One full rotation equals 360 degrees, a convention originating from ancient Babylonian mathematics. Each degree represents 1/360 of a complete circle. For more precise work, degrees are subdivided into 60 arcminutes (′), and each arcminute into 60 arcseconds (″), following the sexagesimal system.

The radian (symbol rad) is the SI unit of angular measure. One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. Since the circumference of a circle is 2πr, a full rotation equals 2π radians (approximately 6.28318 rad). According to Wikipedia, one radian is approximately 57.2958 degrees. Radians are dimensionless and preferred in calculus and physics because they simplify derivative formulas.

The gradian (or gon, symbol grad) divides a right angle into 100 equal parts, making a full circle 400 gradians. It was introduced during the French Revolution as part of the metric system and remains in use in surveying and some European countries. The turn (or revolution) represents one complete 360° rotation and is useful in mechanical engineering and rotational kinematics.

Here are the most frequently needed angle conversions used in education and professional work:

• 90 degrees = π/2 radians ≈ 1.5708 rad = 100 gradians = 0.25 turns
• 180 degrees = π radians ≈ 3.1416 rad = 200 gradians = 0.5 turns
• 360 degrees = 2π radians ≈ 6.2832 rad = 400 gradians = 1 turn
• 1 degree = 60 arcminutes = 3600 arcseconds ≈ 0.01745 radians
• 1 radian ≈ 57.2958 degrees ≈ 63.662 gradians

These conversions are essential when working across different fields. Engineers in the United States often work in degrees, while physicists and mathematicians worldwide predominantly use radians. Surveyors in continental Europe may encounter gradians on older equipment.

Angle measurement appears in virtually every technical discipline. In astronomy, celestial coordinates are expressed in degrees, arcminutes, and arcseconds. The angular diameter of the Sun and Moon as seen from Earth is about 30 arcminutes, while the resolving power of the Hubble Space Telescope is measured in milliarcseconds.

In navigation and GPS, latitude and longitude are angular measurements. A difference of 1 degree of latitude corresponds to approximately 111 kilometers on Earth's surface. Pilots and mariners must constantly convert between compass bearings (degrees) and mathematical calculations.

In engineering and construction, slope angles determine the safety of roads, ramps, and roofs. The angle of repose for granular materials and the thread angle on screws are critical design parameters. For quick slope calculations, try our slope calculator.

In computer graphics and game development, rotations are typically stored in radians because graphics APIs and physics engines use radian-based trigonometric functions. Animators working with Euler angles or quaternions need to convert freely between degrees and radians. For advanced mathematical computations, our scientific calculator handles trigonometric functions in both modes.