Enter Polar Coordinate Values
Tip: Negative radius values are supported. The calculator also reports the equivalent positive-radius angle for interpretation.
Formula Used
x = r cos(θ) y = r sin(θ) Distance from origin = √(x² + y²) = |r| Reverse angle check = atan2(y, x)Here, r is the radius and θ is the angle. The calculator first converts the chosen angle unit into radians, then evaluates cosine and sine to obtain the Cartesian coordinates.
How to Use This Calculator
- Enter the radius value for your polar coordinate.
- Type the angle and choose degrees, radians, or gradians.
- Set the decimal precision for the displayed results.
- Enable angle normalization when you want the input wrapped into a standard cycle.
- Keep verification enabled to compare the reverse angle from the converted point.
- Press Convert Now to show the result above the form. Use the export buttons to save the output as CSV or PDF.
Example Data Table
| Radius (r) | Angle (θ) | Unit | x | y | Position |
|---|---|---|---|---|---|
| 5 | 53.130102 | Degrees | 3 | 4 | Quadrant I |
| 10 | 0.523599 | Radians | 8.660254 | 5 | Quadrant I |
| 12 | 250 | Gradians | -8.485281 | -8.485281 | Quadrant III |
| -7 | 45 | Degrees | -4.949747 | -4.949747 | Quadrant III |
Frequently Asked Questions
1. What does this calculator convert?
It converts a polar coordinate, written as radius and angle, into its Cartesian form. The output gives x, y, position, distance from origin, and optional reverse-angle verification.
2. Which angle units are supported?
The calculator supports degrees, radians, and gradians. Choose the correct unit before submitting so the trigonometric conversion uses the intended angular scale.
3. Why would I normalize the angle?
Normalization wraps large positive or negative angles into one complete cycle. This helps present a standard reference angle without changing the final Cartesian point.
4. Can the calculator handle a negative radius?
Yes. A negative radius is valid in polar notation. The calculator computes x and y directly and also shows an equivalent positive-radius angle to make interpretation easier.
5. Why does the distance from origin equal |r|?
After conversion, the point’s distance from the origin is √(x² + y²). That value matches the absolute radius because polar coordinates define distance from the origin directly.
6. What is the reverse verification angle?
It is the angle rebuilt from the converted Cartesian point using atan2(y, x). This helps verify that the coordinate pair matches the intended polar direction.
7. When should I use more decimal precision?
Use higher precision for laboratory work, graphing comparisons, simulations, or engineering tasks where rounded cosine and sine values could hide small but important differences.
8. Can I save the result for reports or homework?
Yes. Once the result appears, use the CSV button for spreadsheet-friendly output or the PDF button for a neat downloadable summary.