What is an Angle?

In geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles are a fundamental concept used to describe rotation, slopes, and the relationship between lines and shapes. Measuring angles accurately is essential in countless fields, from astronomy and physics to architecture and computer graphics. While we most commonly think in degrees, several other units are used for specific scientific and mathematical applications.

Common Units of Angular Measure

Our converter handles the most common units used to measure angles.

• Degree (°): This is the most familiar unit of angular measure. A full circle is divided into 360 degrees. This system dates back to the ancient Babylonians, who used a base-60 number system. The number 360 was likely chosen because it is highly divisible and approximates the number of days in a year.
• Radian (rad): The radian is the standard unit of angular measure in the International System of Units (SI) and is used in most areas of mathematics and physics. A radian is defined as the angle subtended at the center of a circle by an arc that is equal in length to the circle's radius. A full circle is equal to 2π radians.
• Gradian (grad): Also known as the gon, the gradian is a unit of plane angle, equivalent to 1/400 of a full circle. This means a right angle is 100 gradians. It was introduced as part of the metric system's effort to decimalize everything, but it never gained widespread popularity outside of some specialized fields like surveying.
• Turn: This is the simplest and most intuitive unit. One turn represents a full rotation, a complete circle. It's often used in geometry, technology (e.g., describing the rotation of a motor), and in sports (e.g., a "1080" in snowboarding is three full turns).

How Unit Conversion Works

All angle conversions are based on their relationship to a full circle.

360° = 2π radians = 400 gradians = 1 turn

Using these equivalences, we can derive the conversion factors. To convert from one unit to another, you first convert the original value to a common base unit (like degrees) and then convert from that base unit to your target unit.

For example, to convert 1 Radian to Degrees:

1 rad × (180° / π rad) ≈ 57.296°

Our calculator stores these conversion factors and performs the calculations for you instantly.

Real-World Applications

• Navigation and Astronomy: Degrees (and their subdivisions, minutes and seconds of arc) are used to plot positions on Earth (latitude and longitude) and in the sky.
• Physics and Engineering: Radians are the standard in these fields for calculating angular velocity, rotational motion, and wave phenomena. The formulas of calculus and physics are much simpler when expressed in radians.
• Computer Graphics and Gaming: Both degrees and radians are used extensively to calculate the rotation of objects in 2D and 3D space.
• Construction and Carpentry: Degrees are used every day to measure angles for cuts, joints, and structural alignments.

Frequently Asked Questions (FAQs)

1. How do I use the Angle Converter?

Simply type a value into any of the four input boxes (Degrees, Radians, Gradians, or Turns). The other three boxes will automatically update with the converted values in real-time. The animated protractor will also adjust to visualize the angle.

2. Why use Radians instead of Degrees?

While degrees are more common in everyday life, radians are the "natural" unit for measuring angles in mathematics. They are based on the fundamental properties of a circle (its radius) rather than an arbitrary division like 360. This makes many important mathematical formulas, especially in calculus and trigonometry, much simpler and more elegant when expressed in radians.