One degree is 0.0174532925 radians, which is π divided by 180. The radian is not just another angle unit — it is the only one that makes calculus work, and every trigonometric function in every programming language expects it.
About the units
A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. That definition makes it dimensionless: it is a length divided by a length, which is why it disappears from equations rather than cluttering them. A full circle is 2π radians because the circumference is 2π times the radius. The degree, by contrast, is arbitrary. Its 360 divisions come from Babylonian astronomy, chosen because 360 is close to the number of days in a year and has 24 divisors, making it easy to split into halves, thirds, quarters, fifths, sixths and so on.
The exact factor
The exact relationship is 1° = π/180 radians = 0.01745329251994329... radians. The reason it matters is what happens in calculus: the derivative of sin(x) is cos(x) only when x is in radians. In degrees the derivative picks up a factor of π/180, and every formula in physics and engineering would carry that constant around. Small-angle approximations work the same way — sin(θ) ≈ θ holds for radians, not degrees. Common values worth knowing: 90° is π/2, 180° is π, 45° is π/4, and 1 radian is about 57.296°.
Where you meet this conversion
Programming above all. The trigonometric functions in JavaScript, Python, C and essentially every other language take radians, and passing degrees is one of the most common bugs in graphics and game code — the result is not an error message but a wrong picture. Physics uses radians throughout for angular velocity, phase and oscillation. Navigation, surveying and everyday geometry use degrees, so any calculation that starts from a bearing and ends in a computation crosses this boundary.