Visual Degrees vs Radians

Plain English first

Both degrees and radians measure angles. They measure the exact same angles — just in different units, the way miles and kilometres both measure distance.

Degrees split a circle into 360 equal slices. Why 360? Ancient calendar convenience. It divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 45, 60, 72, 90, 120, 180. Very divisible, very convenient.

Radians measure angles by how much arc length they produce on a circle with radius 1. One radian = one radius-length of arc. A full circle produces 2π radius-lengths of arc, so a full circle = 2π radians.

Use degrees when talking to humans. Use radians when talking to code or math.

Standard math notation

Full circle:  360° = 2π radians
Half circle:  180° = π radians
Quarter:       90° = π/2 radians

Conversion:
  radians = degrees × π / 180
  degrees = radians × 180 / π

1 radian ≈ 57.296°

Verbose Python with descriptive names

PI = 3.141592653589793

def convert_degrees_to_radians(angle_in_degrees):
    """
    Convert a human-friendly degree angle to radians for use in math.

    A full circle (360°) equals 2π radians.
    So each degree equals π/180 radians.
    """
    radians_per_degree = PI / 180
    angle_in_radians   = angle_in_degrees * radians_per_degree
    return angle_in_radians

def convert_radians_to_degrees(angle_in_radians):
    """
    Convert a radian angle back to degrees for display or user input.

    One radian spans about 57.3°.
    """
    degrees_per_radian = 180 / PI
    angle_in_degrees   = angle_in_radians * degrees_per_radian
    return angle_in_degrees

# Sanity checks
print(convert_degrees_to_radians(180))  # should be ≈ 3.14159 (π)
print(convert_degrees_to_radians(90))   # should be ≈ 1.5708  (π/2)
print(convert_degrees_to_radians(360))  # should be ≈ 6.2832  (2π)
print(convert_radians_to_degrees(PI))   # should be 180.0

Common angle conversions

When to use each

• Taking input from a user: degrees (they are more natural)
• Displaying angles to a user: degrees
• Passing to sin(), cos(), atan2(): radians (almost every language)
• Physics simulations, rotations: radians
• Best practice: convert at the entry point, work in radians throughout

Common mistakes

• Passing degrees to sin() or cos() — these functions expect radians. sin(90) in Python is not 1; sin(π/2) is.
• Multiplying by π/180 in the wrong direction (ending up with a very large number instead of a small one). Radians are smaller: a full circle is ≈6.28, not 360.
• Mixing units inside a calculation without realizing it.

See also

• Visual Radians Without Math Libraries
• Visual Pi Dependencies
• Visual Angle Wrapping