Convert rectangular coordinates
Use x and y values. The calculator returns the radius and direction in radians.
Formula used
The conversion uses the point’s distance from the origin and its quadrant-safe direction.
θ = atan2(y, x)
θ0–2π = θ when θ ≥ 0, otherwise θ + 2π
r is the nonnegative radius. θ is the angle in radians. The atan2 function uses both coordinate signs. It therefore identifies the correct quadrant.
How to use this calculator
- Enter the horizontal x coordinate.
- Enter the vertical y coordinate.
- Choose the desired decimal precision.
- Select your preferred radian angle range.
- Add a unit label when it helps.
- Select Convert to polar.
- Review the radius, angle, quadrant, and reference angle.
- Download a CSV or PDF summary when needed.
Example conversions
These familiar points show how coordinates map to polar values.
| Rectangular point | Radius | Radians, 0 to 2π | Degrees | Quadrant or axis |
|---|---|---|---|---|
| (3, 4) | 5 | 0.9273 | 53.13° | Quadrant I |
| (−3, 4) | 5 | 2.2143 | 126.87° | Quadrant II |
| (−3, −4) | 5 | 4.0689 | 233.13° | Quadrant III |
| (3, −4) | 5 | 5.3559 | 306.87° | Quadrant IV |
| (0, 5) | 5 | 1.5708 | 90.00° | Positive y-axis |
Understanding rectangular and polar coordinates
Rectangular coordinates locate a point with horizontal and vertical movement. The ordered pair uses x and y values. Polar coordinates describe the same point differently. They use a distance and an angle. The distance is the radius. The angle starts at the positive x-axis. Radians are common in trigonometry, engineering, and calculus. They link angle size to circle geometry. One complete turn equals two pi radians. This calculator converts a rectangular point into a polar result. It also supplies checks. These support graphing, homework, technical tasks.
Why atan2 gives reliable direction
A simple inverse tangent uses y divided by x. That approach can hide the correct quadrant. Opposite quadrants may share one tangent value. The atan2 function prevents this issue. It evaluates both coordinates together. It returns an angle matching the point direction. The calculator then uses the chosen angle range. Select zero through two pi for positive rotation. Select negative pi through pi for signed rotation. Both ranges describe the same direction. Atan2 remains useful when x equals zero or becomes negative.
Reading the radius
The radius is the straight-line distance from the origin. Standard polar radius values are never negative. The calculator uses the Pythagorean relationship for this value. Decimal coordinates are supported. Set the precision field for the displayed decimal count. Internal computation still uses normal floating-point math. Display rounding changes only the shown text. It does not alter the coordinate relationship. A point at the origin has radius zero. Its direction is undefined. Every angle reaches the origin from the same location.
Where radians help
Students use polar conversion in trigonometry and coordinate geometry. Engineers use radians in rotation models and signal calculations. Programmers use radians in many mathematical libraries. Designers use polar points for circular paths and repeated layouts. Scientists use them for direction and distance descriptions. The result card includes degrees. The reference angle shows the nearest x-axis direction. The quadrant label adds another check. These details make it easier to confirm the result before entering it elsewhere.
Avoiding input mistakes
Enter signed coordinate values carefully. Negative x moves the point left. Negative y moves the point downward. Do not place coordinate labels inside the number fields. Use a decimal point for fractional values. Check the selected angle range before calculating. Negative angles are valid in the signed range. The same direction can appear positive in another range. Keep precision for sensitive measurements. Review the result card before exporting. Try points in every quadrant. Their signs reveal how angle normalization behaves.
Reusing your calculation
The result provides a polar pair, radius, radians, degrees, quadrant, and reference angle. Copy the values into notes, assignments, reports, or software. Download a CSV record for a spreadsheet. The PDF option makes a summary. Use the example table to compare known points. Repeat a calculation with another range to understand normalization. Conversion gets easier when coordinate signs match direction. This page gives main values and useful checks. Use them to complete work with fewer angle errors.
Frequently asked questions
1. What is rectangular to polar conversion?
It changes an x and y coordinate into a radius and angle. Both coordinate systems identify the same point.
2. Why does this calculator use radians?
Radians are the standard angular unit for many trigonometric, calculus, engineering, and programming formulas. They connect directly with circle measurements.
3. What does atan2 do?
atan2 calculates an angle from y and x together. It preserves sign information, so it can identify the correct quadrant.
4. Why can the same point have two angle forms?
An angle may be shown from zero to two pi or from negative pi to pi. The two forms can point in the same direction.
5. Can I enter decimal coordinates?
Yes. Enter positive or negative decimal values. Choose a suitable displayed precision for the final result.
6. What happens at the origin?
The radius is zero. The angle is undefined because every direction from the origin ends at the same point.
7. Is the radius ever negative?
No. This calculator uses the standard polar convention, where radius is nonnegative and the angle supplies direction.
8. What is a reference angle?
It is the smallest positive angle between the terminal side and the nearest x-axis direction. It ranges from zero to pi over two.
9. Are degrees included with the result?
Yes. Radians remain the main output, while degrees provide a quick comparison. You can choose whether degrees appear in CSV downloads.
10. What unit does the radius use?
The radius uses the same measurement unit as x and y. Add an optional label such as cm, m, or ft.
11. How accurate are the results?
The calculations use standard floating-point mathematics. Choose up to twelve displayed decimal places, then apply suitable rounding for your work.