What is the formula for converting radians to degrees?

Multiply the radian value by 180/π (approximately 57.2958). For example, π/4 radians × (180/π) = 45 degrees. The π terms cancel, making common radian values easy to convert.

How many degrees are in one radian?

One radian equals approximately 57.2958 degrees (exactly 180/π degrees). A radian is the angle subtended at the center of a circle by an arc equal in length to the radius.

Why do mathematicians use radians instead of degrees?

Radians simplify many mathematical formulas. Calculus with trigonometric functions works cleanly in radians (e.g., the derivative of sin(x) is cos(x) only when x is in radians). Radians also connect angle measurement directly to arc length: arc length = radius × angle in radians.

How to Convert Radians to Degrees - Formula, Examples & Chart

The Radians to Degrees Formula

The conversion between radians and degrees uses the mathematical constant pi (π):

Degrees = Radians × (180 / π)

Since π radians equals exactly 180 degrees (a half turn), multiplying any radian measure by 180/π (approximately 57.2958) converts it to degrees. This works because a full circle is 2π radians = 360 degrees.

Step-by-Step Examples

Example 1: Converting π/6 Radians to Degrees

Calculation: (π/6) × (180/π) = 180/6 = 30°.
Result: π/6 radians = 30 degrees.

Example 2: Converting 1 Radian to Degrees

Calculation: 1 × (180/π) = 57.296°.
Result: 1 radian = approximately 57.3 degrees.

Example 3: Converting 2.5 Radians to Degrees

Calculation: 2.5 × 57.2958 = 143.24°.
Result: 2.5 radians = approximately 143.2 degrees.

Common Angles Reference Table

Radians	Exact Expression	Degrees
0	0	0°
0.5236	π/6	30°
0.7854	π/4	45°
1.0472	π/3	60°
1.5708	π/2	90°
3.1416	π	180°
4.7124	3π/2	270°
6.2832	2π	360°

Real-World Applications

Programming and Game Development

Most programming languages (C, Python, JavaScript, Java) use radians in their trigonometric functions (sin, cos, tan). Developers often need to convert user-facing degree inputs to radians for calculations and back to degrees for display.

Physics and Engineering

Angular velocity in physics is measured in radians per second. When communicating results to non-technical audiences, converting to degrees per second or revolutions per minute (RPM) is more intuitive.

Surveying and Navigation

Compass bearings are in degrees, but many calculations in geodesy use radians. Surveyors and navigation systems regularly convert between the two for accurate positioning.

Key Relationships to Remember

Memorize these three equivalences and you can derive any common angle:

π radians = 180° (half turn)
2π radians = 360° (full turn)
1 radian ≈ 57.3°

For instant conversions of any angle, use our angle converter.