Degrees to Radians Converter

How to use the degrees to radians converter

1. Enter an angle in the Degrees field to convert degrees to radians, or use the Radians field for the reverse.
2. The result appears instantly below the input fields as you type.
3. Check the Exact row to see the answer expressed as a fraction of π when applicable.
4. Clear both fields to start a new conversion.

How the conversion works

A full circle is 360 degrees or 2π radians. This means one degree equals π/180 radians, and one radian equals 180/π degrees. To convert degrees to radians, multiply the degree value by π and divide by 180. For example, 60 degrees becomes 60 × π / 180 = π/3 ≈ 1.0472 rad. To go from radians to degrees, multiply by 180 and divide by π.

The tool uses JavaScript's built-in Math.PI constant, which provides about 15 decimal digits of precision. That is more than enough for engineering, physics and everyday math. Because the relationship is strictly linear, there are no edge cases where the formula breaks down for finite inputs.

Common angle reference table

This table lists frequently used angles in both units. Memorizing these speeds up trigonometry work considerably and helps you verify calculator output at a glance.

Why radians matter in math and science

Degrees are convenient for navigation and everyday talk because 360 divides evenly into many parts. Radians, however, are the natural unit for mathematics. Calculus formulas like the derivative of sine, d/dx sin(x) = cos(x), only hold when x is measured in radians. If you use degrees, extra scaling factors appear everywhere and complicate derivations.

In physics, angular velocity is expressed in radians per second. When working with rotational mechanics alongside linear conversions such as foot pounds to nm for torque, keeping angles in radians avoids compounding rounding errors across multiple steps. Most programming languages, including JavaScript, Python and C, expect trigonometric function arguments in radians rather than degrees.

Handling negative angles and large values

Angles can be negative, representing clockwise rotation. Converting -90 degrees gives -π/2 radians. The formula works identically for negative numbers: just preserve the sign during multiplication. There is no need to add 360 first unless your specific application requires a positive-only range.

Values larger than 360 degrees represent multiple full rotations plus a remainder. For instance, 720 degrees equals 4π radians, exactly two full circles. The converter handles arbitrarily large inputs without wrapping them back to a single revolution, since the raw numeric value matters in contexts like cumulative rotation tracking in robotics, animation or CNC machining paths.

Exact fractions of pi

When you convert degrees to radians, the result is often a clean fraction of π. The tool checks whether the degree value divided by 180 simplifies to a rational number with a small denominator. If so, it displays the exact form alongside the decimal approximation.

For example, 135 degrees yields 3π/4. This exact representation is preferred in textbooks and exams because it avoids floating point noise entirely. When converting radians to degrees, the tool performs the inverse check to see if the radian value corresponds to a neat degree measure. It tests denominators up to 12, covering all standard angles encountered in geometry and precalculus coursework.

Worked examples

Each result below was produced by the same code the tool runs, so you can check your own numbers against them.