Article — Radians to Degrees Converter
Radians to degrees converter: angle unit conversion
Radians and degrees both measure angles. The exact conversion: 1 rad = 180/π ≈ 57.2958°, and π rad = 180°. Radians are the SI angle unit and the natural choice for calculus, physics and most engineering work. Degrees survive because 360° divides cleanly into many useful sub-angles.
The radian is defined geometrically: the angle subtended by an arc whose length equals the radius. That makes the formula arc length = r × θ a clean identity when θ is in radians. In degrees the same formula needs a π/180 conversion factor every time — which is why calculus textbooks always work in radians.
Radian and degree basics
A full circle is 2π radians or 360 degrees. A right angle is π/2 radians or 90 degrees. A flat angle is π radians or 180 degrees. Numbers in radians look unfamiliar because π is irrational; numbers in degrees feel friendlier because most are integers. Both encode the same geometric content.
The 360 in "degrees per circle" comes from Babylonian astronomy. The 360-day approximate year, combined with their base-60 number system, made 360 a natural choice for full-rotation count. The factor is arbitrary in the sense that any number could have been chosen, but its divisibility (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15…) makes it practically convenient.
The term "radian" was coined by James Thomson (brother of Lord Kelvin) in 1873 examination questions at Queen's College Belfast. Mathematicians had been using the concept for centuries — Roger Cotes wrote about it in 1714 — but it lacked a snappy name until then.
Radians to degrees formula
The conversion: degrees = radians × 180 / π. Or reversed: radians = degrees × π / 180. The factor 180/π ≈ 57.2957795 is irrational, so any computation past a few decimals carries irrational error. Modern calculators and programming environments handle this internally using high-precision π values.
π/6 rad 30°π/4 rad 45°π/3 rad 60°π/2 rad 90°π rad 180°2π rad 360°Common radian values
Most trigonometry problems involve a small number of special angles. Memorising their radian equivalents pays off: π/6 (30°), π/4 (45°), π/3 (60°), π/2 (90°), 2π/3 (120°), 3π/4 (135°), π (180°), 3π/2 (270°), 2π (360°). The unit circle is built around these — sine and cosine values at each one are well-known.
- 0 rad = 0° (start of unit circle)
- π/6 = 30° (sin = 0.5, cos = √3/2)
- π/4 = 45° (sin = cos = √2/2)
- π/3 = 60° (sin = √3/2, cos = 0.5)
- π/2 = 90° (sin = 1, cos = 0)
- π = 180° (sin = 0, cos = -1)
- 2π = 360° (back to start)
Why calculus prefers radians
The derivative of sin(x) is cos(x) — but only when x is in radians. In degrees the derivative picks up a factor of π/180 every time. Same with Taylor series: sin(x) ≈ x for small x only works in radians. Same with the small-angle approximation that lets pendulums and oscillators have simple closed-form solutions.
This isn't a convention; it's a consequence of the radian being defined by arc length over radius. A degree, by contrast, is a 1/360 slice of a circle — dimensionally fine but mathematically clumsy. Working in degrees within calculus is like running a race in shoes that don't fit: doable, but slow.
Radians in physics and engineering
Angular velocity in physics is always rad/s. Angular acceleration is rad/s². Phase in wave mechanics is in radians. Fourier transforms use angular frequency in radians per second. Mechanical engineering torque calculations, rotational kinematics, vibration analysis — all live in radian space.
Degrees show up at the boundary, in places that talk to humans: navigation headings, structural drawings (specifying a chamfer), the angle a satellite dish points. Internally the engineering tools convert to radians, do the math, and convert back to degrees for display.
If a physics formula has sin or cos without any conversion factor near it, the input is in radians. If you see something like sin(θ × π / 180), the formula is using degrees and converting on the fly.
Radians vs degrees vs gradians
A third angle unit, the gradian (or gon), divides a circle into 400 equal parts. 1 grad = 0.9°. Used mostly in European surveying and on some scientific calculators. Almost nobody else uses it. The point of gradians was decimal compatibility — 100 grad = a right angle — but the scheme never caught on outside specialised domains.
A radian is 180/π ≈ 57.296°. A gradian is 0.9°. A degree is 1°. Convert any to any using the relation 2π rad = 360° = 400 grad.
Radians: programming language gotchas
Every major programming language uses radians for trig functions by default. JavaScript's Math.sin(x) expects x in radians. Python's math.sin(x), C's sin(x), Excel's SIN(x) — all radians. Excel adds DEGREES() and RADIANS() as convenience wrappers, but the underlying function is radian-based.
If you pass degrees by mistake, you'll get a wrong but plausible-looking number. Math.sin(90) returns 0.894 — that's the sine of 90 radians, not 90 degrees. The actual answer for 90° is Math.sin(Math.PI / 2) = 1.
Physical calculators have a DEG/RAD/GRAD mode toggle. Most scientific exams fail students because they entered angles in degrees while the calculator was in radian mode. Always check the mode indicator before trusting a trig result.
Radian conversion mental shortcuts
Rough conversion: multiply radians by 57 to get degrees (real factor 57.296). Divide degrees by 57 for radians. For more precision, remember the four canonical pairs: 30°/0.524 rad, 45°/0.785 rad, 60°/1.047 rad, 90°/1.571 rad. Most workplace conversions interpolate from one of these.
The other trick: π/180 ≈ 0.01745. Degrees × 0.01745 = radians, within 0.01% accuracy. For mental math, treat it as "degrees over 60" — close enough for sanity checks (30° / 60 = 0.5 vs actual 0.524, a 5% miss but in the right ballpark).
For engineering work where small angles matter (lasers, optical alignment, surveying), use milliradians (mrad). One milliradian subtends about 1 m at 1 km range — handy for sighting and beam-divergence calculations. The military uses mil markings on rifle scopes, originally one circular mil = 1/6400 of a full turn, close to but not exactly a milliradian.