Convert polar coordinates to Cartesian x and y or convert x and y to canonical polar form, with degrees, radians, gradians, quadrant checks, and equivalent representations.
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Polar Coordinates Conversion Guide
Polar coordinates describe a point using a radial coordinate and an angle from the positive x-axis. The canonical radius is the nonnegative distance from the origin; a negative radial coordinate points in the opposite direction. This representation is useful for circular motion, rotation, vectors and geometry centered around a point.
Key Formulas
| Conversion | Formula | When to Use It |
|---|---|---|
| Cartesian to polar |
r = โ(xยฒ + yยฒ) ฮธ = atan2(y, x) |
Use when the x- and y-coordinates are known and you want radius and angle. |
| Polar to Cartesian |
x = rcos(ฮธ) y = rsin(ฮธ) |
Use when the radius and angle are known and you want the rectangular coordinates. |
| Degrees and radians |
ฮธrad = ฮธdeg ร (ฯ) / (180) ฮธdeg = ฮธrad ร (180) / (ฯ) |
Use when you need to switch the angle unit used by the calculator or problem. |
| Equivalent polar forms |
(r,ฮธ) equiv (r,ฮธ + 2ฯ k) equiv (- r,ฮธ + ฯ + 2ฯ k), k in mathbbZ |
The same point can be written with many different angles, and even with a negative radius. |
How to Convert a Point
- Find the radius using the distance from the origin.
- Find the angle using the signs of both x and y so the quadrant is correct.
- Choose degrees, radians or gradians as needed. One full turn is 360ยฐ = 2ฯ radians = 400 gradians.
- Remember that multiple polar representations can describe the same point.
Use atan2(y, x) rather than arctan(y/x) to retain both coordinate signs. Normalize its direction to the chosen canonical interval. A point on an axis belongs to that axis, not a quadrant; at the origin the angle is undefined or not unique. See the OpenStax conversion formulas and equivalent polar forms.
Quadrant Guide
| Quadrant | Sign of x | Sign of y | Angle Range |
|---|---|---|---|
| I | Positive | Positive | 0ยฐ < ฮธ < 90ยฐ |
| II | Negative | Positive | 90ยฐ < ฮธ < 180ยฐ |
| III | Negative | Negative | 180ยฐ < ฮธ < 270ยฐ |
| IV | Positive | Negative | 270ยฐ < ฮธ < 360ยฐ |
Quick Conversion Examples
| Cartesian Point | Polar Form | Notes |
|---|---|---|
| (1, 0) | r = 1, ฮธ = 0ยฐ | Point lies on the positive x-axis. |
| (0, 1) | r = 1, ฮธ = 90ยฐ | Point lies on the positive y-axis. |
| (-1, 0) | r = 1, ฮธ = 180ยฐ | Point lies on the negative x-axis. |
| (0, -1) | r = 1, ฮธ = 270ยฐ | Also commonly written as โ90ยฐ. |
| (3, 4) | r = 5, ฮธ โ 53.13ยฐ | A classic right-triangle conversion. |
| (-3, 4) | r = 5, ฮธ โ 126.87ยฐ | Same radius as (3, 4), different quadrant. |
Practical Notes
- At the origin: when both coordinates are exactly zero, the radius is zero and the angle is not unique. The calculator reports the angle as undefined rather than assigning 0ยฐ. Small nonzero coordinates are not classified as the origin by a fixed tolerance.
- Angle conventions vary: this calculator uses 0ยฐ โค ฮธ < 360ยฐ, 0 โค ฮธ < 2ฯ radians or 0 โค ฮธ < 400 gradians. Near an axis, it may show an axis angle plus or minus a small offset to avoid rounding away a nonzero direction. Entered polar angles are limited to one billion turns; reduce larger angles before entry.
- Radius units stay the same: if x and y are in meters, then r is also in meters.
- Polar coordinates are not unique: adding a full turn gives the same point.
- Negative radius is valid: it represents the same point as a positive radius with the angle shifted by half a turn.
Common Questions
- Why use polar coordinates instead of Cartesian coordinates?
- They are often simpler for circles, rotation, angles, vectors, and any problem with radial symmetry.
- Why does the angle sometimes come out negative?
- A negative input angle measures clockwise rotation and is valid. The calculator converts the resulting point to a canonical counterclockwise angle below one full turn; another convention may retain the negative angle.
- Can one point have more than one polar form?
- Yes. The same location can be written with different angles, and it can also be written with a negative radius and a shifted angle.
- What is the most reliable way to find the angle?
- Use the two-coordinate angle function so the result lands in the correct quadrant without manual adjustment.
