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 by its distance from the origin and its direction from the positive x-axis. This is often the most natural way to represent 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.
- Convert the angle to degrees or radians if needed.
- Remember that multiple polar representations can describe the same point.
The angle formula is best written with atan2, not just arctan of y divided by x. A basic arctangent can lose quadrant information, while atan2 uses both coordinates and correctly places the angle in Quadrant I, II, III, or IV.
Quadrant Guide
| Quadrant | Sign of x | Sign of y | Angle Range |
|---|---|---|---|
| I | Positive | Positive | 0ยฐ to 90ยฐ |
| II | Negative | Positive | 90ยฐ to 180ยฐ |
| III | Negative | Negative | 180ยฐ to 270ยฐ |
| IV | Positive | Negative | 270ยฐ to 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 zero, the radius is zero and the angle is not unique.
- Angle conventions vary: some results are shown from 0ยฐ to 360ยฐ, while others use negative angles for clockwise rotation.
- 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 angle is still correct. It simply means the direction was measured clockwise instead of counterclockwise.
- 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.
