How this instrument works
A radian is defined by arc and radius rather than by dividing a circle into a round number of parts: sweep out an arc exactly as long as the circle's own radius, and the angle it subtends is one radian. A full turn holds 2π of them, so 360 degrees equals 2π radians, and dividing through gives 1 degree = π⁄180 radian, roughly 0.0174532925199 — a ratio fixed by geometry, not by any committee.
Degrees were never the only alternative to the radian. During the decimalising fever of the 1790s French Revolution, surveyors alongside the newly minted metre also proposed the grade, or gon: a right angle split into 100 parts rather than 90, giving 400 gon to a full turn. It never displaced the degree in everyday life, but it survives on scientific calculators' DEG/RAD/GRAD switch and in some European civil-engineering total stations, a reminder that 360 was always a cultural choice rather than a mathematical necessity.
Radians earn their keep wherever motion gets computed rather than just drawn. Robot-arm inverse kinematics, PLC and servo-drive motion function blocks, wind-turbine yaw and pitch controllers, and the ROS and MoveIt libraries behind most industrial and research robots all work internally in radians, because angular velocity, angular acceleration and arc length only come out clean — ω = Δθ⁄Δt, s = rθ — when θ is measured that way. Degrees stay on the HMI screen and the CAD drawing for a human operator to read.
- Type your angle into the Degrees (deg) field — it opens on 90, a quarter turn.
- Read the Radians (rad) line beneath it; it recalculates on every keystroke.
- Feeding a motion controller that wants radians per second? Divide your radian figure by the move time in seconds to get angular velocity.
- Only angles of zero or more are accepted here; for a negative bearing or rotation, add 360° before entering it.
- Going the other way, multiply your radian figure by 57.2957795131 to recover degrees.
Worked example — a 90° robot-joint move in radians
A six-axis robot arm needs its wrist joint to sweep a clean quarter turn, 90°, between two waypoints in a pick-and-place routine. The trajectory planner inside the controller — whether that is a proprietary servo-drive firmware or a ROS motion-planning stack — accepts joint targets only in radians. Leave Degrees (deg) at its default of 90 and Radians (rad) reads 1.57079632679, exactly π⁄2.
Say that move has to finish in half a second for the cycle time the line requires. Average angular velocity is displacement over time: 1.57079632679 rad ÷ 0.5 s = 3.14159265358 rad/s, which is π rad/s almost to the last digit — a tidy check that the radian conversion, and the motion-planning maths built on it, both landed where geometry says they should.
Questions
Why do robotics and motion-control systems compute in radians instead of degrees?
Because the core equations of rotational motion only stay clean in radians. Angular velocity is ω = Δθ⁄Δt and arc length is s = rθ, both true without any extra scaling factor precisely because a radian is defined as arc length divided by radius — a ratio of two lengths, not an arbitrary slice of a circle. Express θ in degrees instead and both formulas pick up a stray π⁄180 that most motion-planning libraries would rather not carry through every calculation.
Is 0.0174532925199 an exact conversion factor?
No — nothing decimal ever could be. The true multiplier is π⁄180, and π never terminates, so 0.0174532925199 is that ratio cut at twelve significant figures, short by about 2.5 parts in a trillion. Spread across a shaft encoder reporting millions of counts per revolution, that rounding is still many orders of magnitude below anything the hardware itself can resolve.
What is a gradian, and does anything still use it?
A gradian, or gon, splits a right angle into 100 parts rather than 90, giving 400 gon to a full turn — a decimalised alternative proposed alongside the metre during the 1790s French Revolution. It never caught on for everyday use, but it lives on as the GRAD setting on scientific calculators and appears in some continental European surveying total stations, where 100 gon reads as a clean right angle.
How is angular velocity in radians per second related to RPM?
Multiply revolutions per minute by 2π and divide by 60: angular velocity in rad/s equals RPM times 2π over 60. A motor spinning at 1,800 RPM is turning at about 188.5 rad/s. The reverse works the same way in mirror: divide rad/s by 2π and multiply by 60 to recover RPM, a conversion that motor datasheets and servo-drive tuning software both lean on constantly.
Can I enter angles larger than 360° or use this for multiple rotations?
Yes — the multiplication does not wrap values into a single turn. Entering 720°, two full rotations, returns 12.5663706 rad rather than looping back to zero, which matters for anything tracking cumulative shaft rotation, like a stepper motor counting steps across several turns rather than just its position within one.
Why won't this tool accept a negative degree value?
This particular sheet only accepts angles of zero or more, so a negative bearing or a clockwise rotation needs converting first — add 360° to bring it back into range, or simply track the sign separately and apply it to the radian result afterward, since the conversion factor itself is unaffected by direction.
Is a radian the same kind of unit as a steradian?
Related, but one dimension up. A radian measures a flat, two-dimensional angle as arc length over radius; a steradian measures a three-dimensional solid angle as spherical surface area over radius squared, with 4π steradians covering an entire sphere. Steradians show up in lens and LED light-output specifications and antenna gain patterns, wherever a beam spreads across an area rather than sweeping along an arc.