How to Convert Degrees to Radians (and Back)

What Is a Degree?

A degree (°) divides a full circle into 360 equal parts. The choice of 360 goes back to ancient Babylon, which used a base-60 number system. 360 has many divisors (2, 3, 4, 5, 6, 8, 9, 10, 12…), making it easy to divide circles into equal slices without fractions.

Degrees are used in everyday geometry, geography (latitude and longitude), construction, navigation, and most school mathematics. A right angle is 90°, a straight line is 180°, and a full rotation is 360°.

What Is a Radian?

A radian (rad) is the SI unit of angle. One radian is the angle at the center of a circle subtended by an arc equal in length to the radius. Because the circumference of a circle is 2π times the radius, a full circle contains exactly 2π radians — approximately 6.2832 rad.

Radians make mathematical formulas simpler. The derivative of sin(x) is cos(x) only when x is in radians. Arc length equals radius × angle in radians. In physics and engineering, radians simplify wave equations, rotational motion, and signal processing. Most programming languages and calculators use radians by default in trigonometric functions.

The Conversion Formula

The relationship between degrees and radians comes from the full circle: 360° = 2π radians, so 180° = π radians.

To convert degrees to radians: multiply by π/180 (approximately 0.017453).

Example: 90° × π/180 = π/2 ≈ 1.5708 radians.

To convert radians to degrees: multiply by 180/π (approximately 57.2958).

Example: 1 radian × 180/π ≈ 57.296°.

A quick check: 180° = π ≈ 3.14159 rad. If your result is close to this relationship, you are on the right track.

Common Angle Reference Values

These are the angle values you will see most often in trigonometry and engineering:

• 0° = 0 radians
• 30° = π/6 ≈ 0.5236 radians
• 45° = π/4 ≈ 0.7854 radians
• 60° = π/3 ≈ 1.0472 radians
• 90° = π/2 ≈ 1.5708 radians
• 120° = 2π/3 ≈ 2.0944 radians
• 180° = π ≈ 3.14159 radians
• 270° = 3π/2 ≈ 4.7124 radians
• 360° = 2π ≈ 6.2832 radians

Real-World Examples

Programming: JavaScript's Math.sin(), Math.cos(), and Math.atan2() all use radians. To compute sin(45°) in code, you write Math.sin(45 * Math.PI / 180). Forgetting to convert is one of the most common trigonometry bugs.

Navigation and maps: GPS coordinates use decimal degrees. Bearing angles (compass headings) use degrees. When computing great-circle distances between two points on Earth, the formulas internally convert to radians.

Physics and engineering: Angular velocity is measured in radians per second (rad/s). A motor spinning at 3,000 RPM rotates at 3,000 × 2π/60 ≈ 314.16 rad/s.

Construction and carpentry: Angles in blueprints, roof pitches, and miter cuts are nearly always expressed in degrees. Radians appear only in structural engineering calculations.

What About Gradians?

A third unit — the gradian (grad or gon) — divides a full circle into 400 parts. A right angle is exactly 100 gradians. Gradians are used in surveying and some European engineering traditions because they simplify decimal calculations for right-angle geometry.

To convert degrees to gradians: multiply by 10/9. For example, 90° × 10/9 = 100 grad.

Gradians rarely appear outside of surveying and specialized technical fields. For most purposes, you only need degrees and radians.

Conclusion

The core formula is simple: degrees × π/180 = radians, and radians × 180/π = degrees. Memorize the key reference values (90° = π/2, 180° = π, 360° = 2π) and you can convert most common angles in your head. Use our free angle converter for any value instantly.