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Degrees (°) divide a full circle into 360 parts — the everyday, intuitive unit, historically tied to ancient Babylonian base-60 mathematics. Radians (rad) are the natural mathematical unit, defined so that an angle's radian measure equals the arc length it sweeps out on a unit circle divided by the radius — a full circle is exactly 2π radians. Radians are the required unit in calculus and most trigonometric formulas in physics and engineering (including AC phase angle calculations), because derivatives and series expansions of sine/cosine only take their simple textbook forms when the angle is in radians. Gradians (grad), also called "gons," divide a full circle into 400 parts, making a right angle exactly 100 grad — a decimal-friendly unit occasionally used in surveying and some European engineering contexts.
| Unit | Full circle | Right angle |
|---|---|---|
| Degrees | 360° | 90° |
| Radians | 2π rad≈6.2832 | π/2 rad≈1.5708 |
| Gradians | 400 grad | 100 grad |
Multiply by π/180: e.g. 90°×π/180=π/2≈1.5708 radians.
Multiply by 180/π: e.g. 1 radian×180/π≈57.30°.
Most trigonometric functions in programming libraries (sin, cos, tan) and mathematical formulas in calculus/physics are defined to work naturally in radians, since radian measure is what makes derivatives of trig functions and series expansions like the Taylor series take their simplest textbook form.
A gradian (or "gon") divides a full circle into 400 equal parts instead of 360, making a right angle exactly 100 gradians. It sees occasional use in surveying and some European engineering instruments, valued for its clean decimal relationship to right angles, but is far less common than degrees or radians.
In AC circuits, voltage and current can be shifted in time relative to each other (due to inductance or capacitance), and this time shift is expressed as a phase angle — used directly in power factor, impedance, and power triangle calculations throughout AC circuit analysis.
π radians = 180° exactly — this is one of the most useful reference points to memorize, since it makes converting many common angles (90°=π/2, 45°=π/4, 60°=π/3, etc.) quick to do from memory.
2π radians (approximately 6.2832), corresponding to 360° — this comes directly from the radian's definition as arc length divided by radius, and a full circle's circumference is 2π times its radius.
Both are used depending on context: mechanical degrees describe the physical rotor position (0-360° per physical revolution), while electrical degrees describe the phase of the generated waveform, which equals mechanical degrees multiplied by the number of magnetic pole pairs — an important distinction in multi-pole machine analysis.
Match the precision needed by your downstream calculation — control systems and precision positioning often need several decimal places of radian precision, while general conceptual work is usually fine with 1-2 decimal places.
A rough rule of thumb: 1 radian ≈ 57.3°, so dividing a degree value by about 57.3 gives an approximate radian value — useful for quick mental sanity checks, though not a substitute for the exact π/180 conversion in real calculations.
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