Convert a rectangular point
Enter Cartesian coordinates, then choose your preferred angle presentation.
Formula used
The radius measures the point’s straight-line distance from the origin. The angle shows its direction from the positive x-axis.
The atan2 function uses both coordinate signs. It therefore places the angle in the correct quadrant without manual correction.
How to use this calculator
- Enter the x coordinate from your point.
- Enter the y coordinate from your point.
- Choose degrees or radians for the output.
- Select a positive or signed angle range.
- Set decimal places and result notation.
- Press Convert coordinates to view the result.
- Download CSV or PDF after reviewing values.
Example conversions
| Rectangular point | Radius r | Angle θ | Position |
|---|---|---|---|
| (3, 4) | 5 | 53.1301° | Quadrant I |
| (−5, 5) | 7.0711 | 135° | Quadrant II |
| (−6, −8) | 10 | 233.1301° | Quadrant III |
| (4, −4) | 5.6569 | 315° | Quadrant IV |
| (0, 7) | 7 | 90° | Positive y-axis |
Understanding rectangular and polar coordinates
Two ways to locate one point
Rectangular coordinates describe a point with horizontal and vertical movement. The pair uses x first and y second. Polar coordinates describe the same point through distance and direction. The pair uses radius first and angle second. Both systems are useful. The best choice depends on the problem.
Why the radius matters
The radius is always zero or positive. It tells you how far the point is from the origin. The calculator applies the Pythagorean theorem to find it. Squaring removes negative signs. Adding the squares gives a positive distance measure. The square root restores the original measurement scale.
How the angle is found
The angle starts from the positive x-axis. Counterclockwise movement is positive. Clockwise movement is negative in signed form. A basic inverse tangent can find a reference angle. However, it cannot reliably identify every quadrant. This calculator uses atan2. That method checks x and y together.
Choosing degrees or radians
Degrees are familiar for most diagrams. One complete rotation equals 360 degrees. Radians are common in calculus, physics, and programming. One complete rotation equals 2π radians. Either unit describes the same direction. Choose the unit required by your assignment, application, or technical standard.
Positive and signed angle ranges
Positive ranges keep angles above zero. They use 0 through 360 degrees. Signed ranges allow clockwise directions to remain negative. They commonly use negative 180 through positive 180 degrees. The calculator can present either version. This makes comparison with textbook answers much easier.
Handling axes and the origin
Points on an axis have simple angles. Positive x is zero degrees. Positive y is 90 degrees. Negative x is 180 degrees. Negative y is 270 degrees in positive notation. The origin is special. Its radius is zero, so direction has no unique meaning. The calculator displays zero by convention.
Using results in practical work
Polar coordinates help when direction matters. Surveying, robotics, navigation, signals, and engineering use them often. Convert values before plotting vectors or analyzing circular motion. Check the quadrant carefully. A correct radius with a wrong angle points somewhere else. Save your output as CSV or PDF when documentation is needed.
Checking your conversion
Begin by checking the signs of both coordinates. Positive and negative signs reveal the quadrant. Next, estimate the direction before accepting the angle. A point near the x-axis should have a small angle. A point near the y-axis should have an angle near 90 degrees. Then inspect the radius. It must never be negative. Larger coordinate values usually create a larger radius. Finally, compare degree and radian displays carefully. They look different but indicate the same direction. Rounding should happen after calculation, not before. This protects accuracy when values contain many decimal places. When a result seems surprising, plot the point roughly and confirm distance and direction visually on paper.
Frequently asked questions
1. What does rectangular coordinate mean?
A rectangular coordinate uses an ordered pair, written as (x, y). The x value measures horizontal position. The y value measures vertical position. Both values are measured from the origin.
2. What does a polar coordinate contain?
A polar coordinate contains a radius and an angle. The radius gives distance from the origin. The angle gives the direction from the positive x-axis. It is commonly written as (r, θ).
3. Why does this calculator use atan2?
atan2 uses both x and y values. It identifies the correct quadrant automatically. A simple inverse tangent only uses a ratio. That can produce an angle in the wrong quadrant.
4. Can the radius be negative?
This calculator reports a nonnegative radius. That is the standard form. Some advanced conventions allow negative radii, but they require an angle shifted by 180 degrees. Standard positive radius output is easier to compare and verify.
5. What happens when x and y are both zero?
The point is the origin. Its radius is zero. Every direction reaches the origin, so its angle is not unique. This calculator displays zero as a practical convention.
6. Which angle range should I select?
Select the positive range for 0 to 360 degrees or 0 to 2π radians. Select the signed range when your work expects clockwise angles as negative values. Follow your course or project convention.
7. Are degrees and radians interchangeable?
They express the same direction using different scales. A full turn is 360 degrees or 2π radians. Convert only when required, and keep the unit label visible to avoid mistakes.
8. Can I use decimal coordinate values?
Yes. The calculator accepts positive, negative, whole, and decimal values. It calculates the radius and angle from the supplied precision, then formats results using your chosen decimal places.
9. How is the quadrant determined?
The signs of x and y determine the quadrant. Positive x and positive y form Quadrant I. Negative x and positive y form Quadrant II. The other sign combinations form Quadrants III and IV.
10. Why might my answer differ from a textbook answer?
Your textbook may use a different angle range, unit, or rounding rule. Equivalent angles can differ by a full turn. Compare the radius, unit, selected range, and decimal precision before deciding an answer is wrong.
11. What can I download after calculating?
You can download a CSV file containing the inputs and calculated values. You can also create a PDF summary. These options help with reporting, checking calculations, and saving records.