Convert Radians (rad) to Degrees (°)

What Is a Radian?

The radian (rad) is the SI derived unit of plane angle. It is defined as the angle subtended at the center of a circle by an arc equal in length to the radius. A full circle equals 2π radians (≈ 6.2832 rad).

Although mathematically dimensionless, the radian is retained as a named SI unit for clarity in physics, calculus, and engineering. Scientific calculators, programming languages, and spreadsheet functions return inverse-trig results in radians by default.

Key equivalences: 1 rad ≈ 57.2958°; π rad = 180°; 2π rad = 360° = 1 turn.

What Is a Degree?

The degree (°) is a non-SI unit of plane angle defined as exactly 1/360 of a full rotation. It is accepted for use with the SI due to its long-standing importance in navigation, surveying, astronomy, and daily measurement.

The 360-degree circle originates from ancient Babylonian astronomy. Today degrees remain the most intuitive angular unit for human communication — compass headings, map bearings, protractor readings, and GPS coordinates all use degrees.

Key equivalences: 1° = π/180 rad = 60′ = 3,600″ = 10/9 grad.

Radians to Degrees Formula

The conversion factor is exact:

degrees = radians × 180/π exact

This is the inverse of the degree-to-radian formula. Since π rad = 180° by definition, multiplying any radian value by 180/π (≈ 57.2958) gives the equivalent in degrees.

Worked Example

Convert 1 radian to degrees:

1. Multiply: 1 × 180/π = 57.2958°

So 1 radian is about 57.3° — slightly less than one-sixth of a full circle. This is the angle where the arc length equals the radius.

Quick Estimation

For mental math, multiply radians by 57.3. Example: 2.5 rad × 57.3 ≈ 143° (exact: 143.24°). For π-based values, use the identity π = 180° directly: 3π/4 = 3 × 180/4 = 135°.

Common Conversions at a Glance

Which Is Bigger: Radian or Degree?

A radian is about 57.3 times larger than a degree. One radian spans roughly a sixth of a circle, while a degree is a tiny 1/360 slice. This is why radian values for practical angles are small numbers (typically 0 to 2π), while the same angles in degrees range from 0 to 360.

When You Need This Conversion

• Interpreting calculation results: Functions like atan2(), asin(), and acos() in Python, JavaScript, and C return radians. Converting to degrees makes results immediately understandable: atan2(1,1) = 0.7854 rad = 45°.
• Education and exams: Students working through calculus problems convert final answers back to degrees when problems specify degree-based answers or for presentation clarity.
• Engineering specifications: Engineers compute in radians but report in degrees on blueprints and technical drawings. An angular tolerance of 0.0175 rad is more clearly communicated as 1°.
• Robotics and sensor data: Gyroscopes and IMUs output in rad/s. Accumulated rotation of 4.712 rad is more intuitive as 270° — "three-quarter turn" — when monitoring joint angles.