Radian Degree Converter

Angles can be measured in several ways, but the two most commonly encountered units in mathematics, science, and engineering are degrees and radians. While degrees are familiar from everyday life — a right angle is 90°, a full rotation is 360° — radians are the mathematically natural unit and are used almost exclusively in calculus, physics, and most scientific computing. Understanding how to convert between these two systems is a fundamental skill for students and professionals alike.

What Is a Degree?

A degree (°) is defined as 1/360 of a full rotation. The number 360 traces back to ancient Babylonian astronomy, where the year was approximated as 360 days, and the Babylonian number system was base 60 (sexagesimal). This made 360 a convenient choice because it is divisible by many integers: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180.

Degrees are split further into arcminutes (1° = 60 arcminutes) and arcseconds (1 arcminute = 60 arcseconds), which are used in navigation, astronomy, and geodesy for expressing very precise angular measurements. For everyday use, degrees are intuitive and easy to communicate, which is why they remain the standard in navigation, architecture, and general education.

What Is a Radian?

A radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius. In other words, if you take a circle of radius r and walk along the circumference a distance equal to r, the angle you have swept out at the center is exactly one radian. Because a full circle has circumference 2πr, a full rotation equals exactly 2π radians.

Radians make the relationship between angle and arc length perfectly clean: arc length = radius × angle in radians. This elegance extends throughout mathematics. The derivative of sin(x) is cos(x) only when x is measured in radians. Taylor series expansions for trigonometric functions assume radian input. Angular velocity in physics (ω = dθ/dt) is expressed in radians per second. As a result, radians are the SI unit for angle, and virtually all scientific software uses radians by default.

The Conversion Formula

The conversion between degrees and radians follows from the fact that 360° equals 2π radians, which simplifies to 180° = π radians. From this relationship, the conversion formulas are derived:

To convert degrees to radians: multiply the degree value by π/180. For example, 90° × (π/180) = π/2 ≈ 1.5708 radians.

To convert radians to degrees: multiply the radian value by 180/π. For example, π/3 × (180/π) = 60°.

These formulas work for any angle, positive or negative, whether it represents a fraction of a turn or multiple full rotations. Negative angles indicate rotation in the clockwise direction (in the standard mathematical convention).

Common Angle Reference

Certain angles appear so frequently in trigonometry, geometry, and physics that memorizing their radian equivalents is practical. The most important ones are: 0° = 0 rad, 30° = π/6 ≈ 0.5236 rad, 45° = π/4 ≈ 0.7854 rad, 60° = π/3 ≈ 1.0472 rad, 90° = π/2 ≈ 1.5708 rad, 120° = 2π/3 ≈ 2.0944 rad, 135° = 3π/4 ≈ 2.3562 rad, 150° = 5π/6 ≈ 2.6180 rad, 180° = π ≈ 3.1416 rad, 270° = 3π/2 ≈ 4.7124 rad, and 360° = 2π ≈ 6.2832 rad.

The angles 30°, 45°, and 60° are particularly important because they arise from the two special right triangles: the 30-60-90 triangle (with side ratios 1 : √3 : 2) and the 45-45-90 triangle (with side ratios 1 : 1 : √2). These triangles give exact, simple values for all six trigonometric functions and appear constantly in both pure and applied mathematics.

When to Use Degrees vs. Radians

The choice of unit depends on the context. Degrees are preferred in everyday measurements, navigation, construction, and surveying. When communicating with a general audience, degrees are more intuitive — most people have an immediate sense of what a 45° angle looks like.

Radians should be used in any mathematical or scientific context involving calculus, differential equations, or physics. Programming languages and scientific libraries (Python's math module, JavaScript's Math object, C's math.h, MATLAB, NumPy, and virtually all others) take angles in radians for their trigonometric functions. When computing sin(x) in code, x must be in radians. Feeding degrees directly without conversion is one of the most common sources of bugs in scientific programs.

In physics, angular frequency (ω), angular momentum, and rotational kinematics all use radians natively. The formula for the period of a pendulum, the equations of circular motion, and the wave equation all involve angles measured in radians. Similarly, in electrical engineering, the phase of an AC signal is expressed in radians.

Radians in Higher Mathematics

The naturalness of radians becomes even clearer in higher mathematics. The limit lim(x→0) sin(x)/x = 1 holds only when x is in radians. The Maclaurin series sin(x) = x − x³/3! + x⁵/5! − ... and cos(x) = 1 − x²/2! + x⁴/4! − ... are valid only for radian input. Euler's famous identity e^(iπ) + 1 = 0 uses radians. Complex exponentials, Fourier analysis, and signal processing all build on these foundations.

Solid angles, the three-dimensional extension of radians, are measured in steradians (sr). A full sphere subtends 4π steradians at its center, the same way a full circle subtends 2π radians. This parallel reinforces why radians are the natural angular unit.