Polar Coordinates
Convert between (x, y) and (r, θ) with the quadrant handled properly, in degrees or radians.
distance from the origin, never negative here
126.8699° · principal value in (−π, π]
quadrant II
The angle, four ways
Other names for the same point
- (5, 2.214297)
- (−5, −0.927295)
Adding 2π to θ, or negating r and adding π, lands on the same place.
x from -3.22equal scale on both axesto 1.62
Polar coordinates name a point by how far out and which way round instead of by across and up. The distance is r = √(x² + y²); the angle needs care, because arctan(y/x) cannot tell (−3, −4) from (3, 4).
The conversion that is not the hard part
Going from polar to rectangular is unambiguous: x = r cos θ and y = r sin θ, and there is nothing to get wrong. The other direction is where marks are lost. Finding r is just Pythagoras, but the angle is usually taught as θ = arctan(y ÷ x), and that formula throws away information.
The reason is that arctan only ever returns an angle between −90° and 90°, so it cannot reach quadrants II and III at all. Worse, y ÷ x is the same number for a point and its opposite: for (3, 4) and (−3, −4) the ratio is 4/3 either way, so arctan hands back 53.13° for both, when the second point is at −126.87°. The fix is to look at the signs of x and y separately, which is exactly what the two-argument arctan — atan2 — does, and what this page uses.
A point has infinitely many names
Rectangular coordinates are unique; polar ones are not. Adding a full turn to θ lands on the same place, so (5, 53.13°) and (5, 413.13°) are the same point. So is (−5, 233.13°), since a negative r means going backwards along the ray. Courses usually ask for the principal angle, between −π and π, but a question may want the 0-to-2π version or the negative-r form, so the tool lists them.
atan2 uses the signs of x and y separately, so it keeps the quadrant
- 1 Find r with Pythagoras. For (−3, 4): r = √(9 + 16) = 5. The distance is never negative.
- 2 Note which quadrant the point is in. Negative x with positive y puts (−3, 4) in quadrant II, so the angle is between 90° and 180°.
- 3 Take the reference angle. arctan(|y| ÷ |x|) = arctan(4/3) = 53.13°, measured from the nearest part of the x-axis.
- 4 Adjust it for the quadrant. In quadrant II the angle is 180° − 53.13° = 126.87°. Quoting the raw arctan of −53.13° would put the point in the wrong place entirely.
- 5 Give the form the question asks for. (5, 126.87°), or (5, 2.2143 rad), or the 0-to-2π and negative-r versions of the same point.
Why the quadrant has to be checked
All four points have r = 5 and the same |y ÷ x|, so arctan alone cannot separate them.
| Point | Quadrant | Correct θ | arctan(y ÷ x) says |
|---|---|---|---|
| (3, 4) | I | 53.13° | 53.13° — correct |
| (−3, 4) | II | 126.87° | −53.13° — wrong by 180° |
| (−3, −4) | III | −126.87° | 53.13° — wrong by 180° |
| (3, −4) | IV | −53.13° | −53.13° — correct |
When polar is the easier description
Anything built around a centre is simpler in polar form. A circle of radius 3 is just r = 3, against x² + y² = 9. Spirals, roses and cardioids have short polar equations and awkward rectangular ones, which is why they are always introduced this way. The same holds in physics: orbits, radiation patterns and anything with rotational symmetry are naturally written from the centre outwards.
Two conventions to watch. The angle is measured anticlockwise from the positive x-axis in mathematics, but navigation and surveying measure clockwise from north, so a bearing of 30° is not the same as a polar angle of 30°. And the origin is genuinely special: at r = 0 the angle carries no information, because every θ names the same point.
On exact values, the tool reports θ as a multiple of π whenever it is a simple one — π/4 rather than 0.785398 — because that is the form a marking scheme usually wants. Where the angle is not a neat fraction of π it gives the decimal instead, rather than forcing a form that would be false.