Calculator Form
This page uses a single-column page flow, while the form fields adapt into 3, 2, or 1 columns by screen width.
Example Data Table
These sample values help you verify expected radius and angle outputs.
| X | Y | Radius r | Angle θ (Degrees) | Location |
|---|---|---|---|---|
| 3 | 4 | 5.0000 | 53.1301° | Quadrant I |
| -4 | 4 | 5.6569 | 135.0000° | Quadrant II |
| -5 | -12 | 13.0000 | -112.6199° | Quadrant III |
| 0 | 7 | 7.0000 | 90.0000° | Positive Y-Axis |
| 8 | 0 | 8.0000 | 0.0000° | Positive X-Axis |
Formula Used
Rectangular coordinates are written as (x, y). Polar coordinates are written as (r, θ).
Radius formula: r = √(x² + y²)
Angle formula: θ = atan2(y, x)
The atan2 function is preferred because it identifies the correct quadrant automatically. That makes the output more reliable than using a basic inverse tangent alone.
Angle conversion rules are:
- Degrees = Radians × 180 / π
- Gradians = Radians × 200 / π
An equivalent alternate polar form also exists for nonzero points: (-r, θ + π).
How to Use This Calculator
- Enter the rectangular x coordinate.
- Enter the rectangular y coordinate.
- Select your preferred angle output unit.
- Choose the angle range style you want.
- Set the decimal precision for the output.
- Decide whether to show steps and alternate form.
- Press Convert Coordinates to generate the result.
- Review the graph, detailed outputs, and downloadable files.
FAQs
1. What does this calculator convert?
It converts rectangular coordinates (x, y) into polar coordinates (r, θ). It also shows radius, angle in multiple units, quadrant information, and an alternate equivalent polar pair.
2. Why is atan2 used instead of tan-1(y/x)?
The atan2 function detects the correct quadrant automatically. A simple inverse tangent can lose sign information and produce the wrong angle when x is negative or zero.
3. What happens when the point is at the origin?
When x = 0 and y = 0, the radius becomes zero. The angle is not uniquely defined there, so the calculator marks some angle-based outputs as undefined.
4. What is the alternate polar pair?
Any nonzero polar point can be written in more than one way. For example, (r, θ) is equivalent to (-r, θ + π), or the same shift in your chosen unit.
5. Which angle range should I choose?
Use the signed range when you want positive and negative angles. Use the full range when you prefer angles from zero through one complete rotation.
6. Can this calculator handle decimals and negative values?
Yes. It accepts positive numbers, negative numbers, and decimals for both coordinates. That makes it useful for geometry, vectors, navigation, and engineering practice.
7. What does the plotted graph show?
The graph shows the coordinate point on the plane and a line from the origin to that point. This visually represents the radius used in the polar conversion.
8. What do the CSV and PDF downloads include?
They include the current calculation summary, such as x, y, radius, angle values, location, and related outputs. This helps with records, homework checking, or sharing results.