How this instrument works
A rectangular coordinate pair (x, y) and a polar pair (r, θ) describe the identical point two different ways. Picture the segment from the origin to (x, y): its length is r, and its direction, measured counterclockwise from the positive x-axis, is θ. Because that segment is the hypotenuse of a right triangle with legs x and y, r falls straight out of the Pythagorean relation, r = √(x² + y²). The angle takes a second, separate calculation, because length and direction are genuinely independent pieces of information about the same segment.
Finding θ is where a naive approach breaks. The ratio y/x alone cannot tell (3, 4) apart from (−3, −4): both give y/x = 4/3, so plain arctan(y/x) returns the same 53.13° for either point, even though one sits in the first quadrant and the other in the third. The fix is atan2(y, x), a function that takes the two coordinates separately rather than pre-dividing them, so the individual signs of x and y survive into the answer and the correct quadrant comes out automatically. The two-argument arctangent traces back to 1960s Fortran math libraries and is now standard in essentially every programming language, always in this y-then-x argument order.
One case sits outside the whole scheme: at the origin, x = 0 and y = 0 give r = 0, but a point with no length has no direction to speak of, so θ is conventionally left at 0 rather than genuinely defined. Elsewhere, remember that direction repeats every full turn — adding 360° to θ names the same ray — so atan2 fixes one representative value in (−180°, 180°] rather than reporting every equivalent angle.
- Enter the point's horizontal offset from the origin in x and its vertical offset in y; either one can be negative.
- Read r (radius) for the straight-line distance from the origin out to that point.
- Read θ (angle) for the direction of that same segment, measured counterclockwise from the positive x-axis.
- Toggle θ's unit between degrees, radians, and turns depending on which convention your next step needs.
- If both x and y are 0, expect r = 0 and treat the reported θ as a placeholder rather than a real bearing.
Worked example — locating the point (3, 4)
Take a point three units right and four units up from the origin: x = 3, y = 4. That is a 3-4-5 right triangle in its native habitat, so r = √(3² + 4²) = √25 = 5 exactly. The direction takes the second formula: θ = atan2(4, 3) = 0.9272952180016122 radians, which the dial reports as 53.130102° once converted, since degrees is the field's default unit.
Now flip both coordinates to (−3, −4). Squaring erases sign, so r is still 5 — but the direction is not the same at all. atan2(−4, −3) returns −2.214297435588181 radians, close to −126.87°, placing the point in the opposite corner of the plane. A shortcut that only computed arctan(y/x) would find atan(4/3) in both cases, the identical 53.13°, and quietly send the second point to the wrong quadrant. Reading x and y separately, the way atan2 does, is what keeps the two points apart.
Questions
How do you convert x, y coordinates into r and θ?
Compute r = √(x² + y²), the straight-line distance from the origin, using the Pythagorean relation on x and y as the two legs. Then compute θ = atan2(y, x), the counterclockwise angle from the positive x-axis, using the two-argument arctangent rather than plain arctan(y/x).
Why use atan2 instead of just arctan(y/x)?
Because y/x alone loses information: (3, 4) and (−3, −4) share the same ratio, 4/3, so arctan(y/x) returns the identical 53.13° for both, even though they sit in opposite quadrants. atan2(y, x) reads the sign of x and y separately, so it always lands the angle in the correct quadrant without any extra sign-checking by hand.
What happens when x is 0?
Plain arctan(y/x) would divide by zero and fail. atan2 handles it directly: atan2(y, 0) returns exactly 90° for positive y and −90° for negative y, since a point straight above or below the origin has a perfectly well-defined direction even though there is no horizontal component to divide by.
What angle does this calculator report for the origin, x = 0 and y = 0?
r comes out as exactly 0, and θ is set to 0 by convention rather than measured, because a point with zero length has no meaningful direction. Treat that particular θ as a placeholder, not a real bearing — it's the one input pair where the polar description genuinely can't do what the rectangular one can.
Is this the same as converting a complex number to polar form?
The arithmetic is identical — r and θ come from the same two formulas either way — but the purpose differs. A complex number's polar form, r·(cos θ + i sin θ), exists so that multiplying two complex numbers becomes multiplying their r values and adding their θ values; a plane point's polar form is just a second way to name a location, with no multiplication rule attached.
Why is the angle limited to between −180° and 180°?
Direction repeats every full turn, so infinitely many angles describe the same ray — 53.13° and 413.13° point the same way. atan2 sidesteps that ambiguity by always returning the one representative value in (−180°, 180°], the smallest signed turn from the positive x-axis to the point.