How to Convert Angles

Converting between angle units is essential in mathematics, navigation, engineering, and programming. This guide covers the most common conversions with clear formulas and practical examples you can apply immediately.

Converting Degrees to Radians

The most common angle conversion in mathematics and programming.

Formula

radians = degrees × (π ÷ 180)

Step-by-Step Process

1. Take the angle in degrees
2. Multiply by π (approximately 3.14159)
3. Divide by 180
4. Simplify if possible

Examples

• 45°: 45 × π/180 = π/4 ≈ 0.785 rad
• 90°: 90 × π/180 = π/2 ≈ 1.571 rad
• 60°: 60 × π/180 = π/3 ≈ 1.047 rad
• 30°: 30 × π/180 = π/6 ≈ 0.524 rad

Converting Radians to Degrees

Formula

degrees = radians × (180 ÷ π)

Step-by-Step Process

1. Take the angle in radians
2. Multiply by 180
3. Divide by π
4. Round as needed

Examples

• π/4 rad: (π/4) × 180/π = 45°
• π/6 rad: (π/6) × 180/π = 30°
• 2 rad: 2 × 180/π ≈ 114.59°
• 1 rad: 1 × 180/π ≈ 57.30°

Converting Degrees to Gradians

Gradians (also called gons) are used in some European surveying.

Formula

gradians = degrees × (10 ÷ 9)

Or: gradians = degrees × 1.1111...

Examples

• 90°: 90 × 10/9 = 100 grad
• 45°: 45 × 10/9 = 50 grad
• 180°: 180 × 10/9 = 200 grad
• 360°: 360 × 10/9 = 400 grad

Converting Gradians to Degrees

Formula

degrees = gradians × (9 ÷ 10)

Or: degrees = gradians × 0.9

Examples

• 100 grad: 100 × 0.9 = 90°
• 50 grad: 50 × 0.9 = 45°
• 200 grad: 200 × 0.9 = 180°
• 400 grad: 400 × 0.9 = 360°

Converting DMS to Decimal Degrees

DMS (degrees, minutes, seconds) is common in navigation and surveying.

Formula

decimal degrees = degrees + (minutes ÷ 60) + (seconds ÷ 3600)

Step-by-Step Process

1. Keep the degrees as-is
2. Divide minutes by 60
3. Divide seconds by 3600
4. Add all three values

Example: Convert 45°30'15" to decimal

• Degrees: 45
• Minutes: 30 ÷ 60 = 0.5
• Seconds: 15 ÷ 3600 = 0.00417
• Total: 45 + 0.5 + 0.00417 = 45.50417°

Converting Decimal Degrees to DMS

Step-by-Step Process

1. The whole number is the degrees
2. Multiply the decimal by 60—the whole number is minutes
3. Multiply remaining decimal by 60—this is seconds

Example: Convert 45.50417° to DMS

• Degrees: 45
• Decimal: 0.50417 × 60 = 30.25 → Minutes: 30
• Decimal: 0.25 × 60 = 15 → Seconds: 15
• Result: 45°30'15"

Common Angle Reference Table

Programming Angle Conversions

JavaScript

const toRadians = degrees => degrees * (Math.PI / 180);

const toDegrees = radians => radians * (180 / Math.PI);

Python

import math

radians = math.radians(degrees)

degrees = math.degrees(radians)

Important Note

Most programming language math libraries (sin, cos, tan) expect radians. Always convert degrees to radians before using these functions.

Conclusion

Angle conversion follows straightforward formulas once you remember the key relationships: π radians = 180 degrees, and 90 degrees = 100 gradians. For DMS conversions, remember that there are 60 minutes per degree and 60 seconds per minute. Practice with common angles like 30°, 45°, 60°, and 90° to build intuition.

Related Articles

• Degrees vs Radians: When to Use Each Angle Unit
• Angle Units Worldwide: Degrees, Radians, Gradians, and More