Degrees to radians

Trigonometry homework and most programming languages' math libraries expect angles in radians, while everyday angle measurement, compass headings, and most non-technical contexts use degrees, so converting between the two comes up constantly in both classroom and code. A full circle is 360 degrees or exactly 2π radians, the anchor behind every degrees-to-radians conversion, and a right angle, 90 degrees, is exactly π divided by 2 radians, a pairing worth memorizing since it recurs constantly in trigonometry.

Game and graphics programming hits the same conversion from a design angle: a 3D engine like Unity or Three.js stores and computes every rotation internally in radians, even though the editor's rotation sliders and inspector panels display degrees for the designer's convenience, so any rotation value typed or read programmatically has to cross this exact conversion.

A full circle is defined as 360 degrees and also as exactly 2π radians, so one degree equals π/180 radians — the irrational constant 0.01745329252.

rad = ° × 0.01745329252

A 45° angle converts to 45 × 0.01745329252 = 0.785398 rad, exactly π/4. A 90° right angle converts to 90 × 0.01745329252 = 1.570796 rad, exactly π/2. A 270° angle converts to 270 × 0.01745329252 = 4.712389 rad, exactly 3π/2.

Going the other way, multiply radians by 57.29577951: a result of 1 radian converts to 1 × 57.29577951 = 57.2958°, the well-known fact that one radian is a little under 57.3 degrees.

Because π is irrational, most radian results don't terminate cleanly, so rounding to 4–6 decimal places is standard practice; for the common "nice" angles — 30°, 45°, 60°, 90°, 180° — it's usually clearer to express the answer as a fraction of π (π/6, π/4, π/3, π/2, π) rather than a long decimal.

The mistake specific to this pair is forgetting that virtually every programming language's built-in trig functions — sin(), cos(), atan2() — expect radians, not degrees, by default. Passing 45 directly into sin(45) expecting "sin of 45 degrees" actually computes the sine of 45 radians, a completely different and much larger angle, producing a nonsensical result unless the degrees are converted first.