Convert Coordinates
Use any finite decimal or whole-number coordinate values.
Example Data Table
| Rectangular pair | Radius r | Angle θ in degrees | Polar pair |
|---|---|---|---|
| (3, 4) | 5 | 53.1301° | (5, 53.1301°) |
| (-3, 4) | 5 | 126.8699° | (5, 126.8699°) |
| (-3, -4) | 5 | 233.1301° | (5, 233.1301°) |
| (0, -6) | 6 | 270° | (6, 270°) |
Formula Used
Radius: r = √(x² + y²)
Angle: θ = atan2(y, x)
Reverse check: x = r cos θ and y = r sin θ
The radius measures distance from the origin. The atan2 function uses both coordinate signs. It therefore selects the correct quadrant. A basic arctangent calculation alone can confuse Quadrants II, III, and IV.
How to Use This Calculator
- Enter the horizontal position in the x-coordinate field.
- Enter the vertical position in the y-coordinate field.
- Choose degrees or radians for the main angle display.
- Select a positive or signed angle range.
- Choose the number of decimal places you need.
- Press Convert to Polar and review the result panel.
- Download a CSV or PDF copy after calculating.
Understanding Rectangular and Polar Coordinates
Two Ways to Describe One Point
Rectangular coordinates describe a point by horizontal and vertical movement. The notation is written as (x, y). Start at the origin. Move along the x-axis. Then move along the y-axis. Polar coordinates use a different description. They use distance and direction. The notation is written as (r, θ). The radius r is the distance from the origin. The angle θ shows the turning direction from the positive x-axis.
Finding the Radius
The radius comes from the Pythagorean theorem. Square x. Square y. Add both values. Then take the square root. A point at (3, 4) gives r = √(9 + 16). The radius is 5. Radius is normally nonnegative. A negative radius is possible in some advanced conventions. This calculator reports the standard positive radius. That makes the result easier to compare and verify.
Choosing the Correct Angle
Angle selection needs more care. The ratio y divided by x can produce a reference angle. It cannot always identify the correct quadrant. For example, (3, 4) and (-3, -4) share the same positive ratio. Yet they point in opposite directions. The atan2 method solves this issue. It reads both signs. The calculator also reports the point location. Use that label to confirm the angle makes geometric sense.
Degrees, Radians, and Equivalent Forms
Degrees are common in geometry classes and coordinate sketches. Radians are common in trigonometry, calculus, and programming. One full turn equals 360 degrees or 2π radians. Both measures describe the same direction. Angles also have equivalent forms. Add 360 degrees or 2π radians to get another valid angle. This converter shows one equivalent angle. It helps you recognize repeated rotations around the origin.
Checks That Improve Accuracy
Review the radius first. It should never be negative in the standard format. Then check the quadrant. Positive x and positive y place a point in Quadrant I. Negative x and positive y place it in Quadrant II. The axes are special cases. Finally, use x = r cos θ and y = r sin θ. Recovered coordinates should match your input after rounding. These checks catch sign errors quickly.
Where This Conversion Helps
Polar coordinates help with circular motion, navigation, graphics, surveying, and complex numbers. They simplify problems involving distance from a center. They also make angles easier to see. Rectangular coordinates remain useful for grids and straight-line work. Knowing both systems gives you flexibility. Convert when direction and radius matter most. Use rectangular form when horizontal and vertical components matter more.
Using Direction and Magnitude
Coordinate conversion also supports vector analysis. A vector can be described by a magnitude and an angle. That is polar language. In engineering, this can simplify force diagrams. In computer graphics, it helps place objects around a center. In navigation, it helps report a bearing and range. Always match the angle convention to your assignment. Some systems measure clockwise angles. Standard mathematics measures counterclockwise angles from the positive x-axis.
Frequently Asked Questions
1. What are rectangular coordinates?
Rectangular coordinates locate a point with horizontal and vertical distances from the origin. They are written as (x, y). The x-value moves left or right. The y-value moves down or up.
2. What are polar coordinates?
Polar coordinates locate a point by its distance from the origin and its direction. They are written as (r, θ). The first value is the radius. The second value is the angle.
3. Which formula finds the radius?
Use r = √(x² + y²). This is the Pythagorean theorem applied to the horizontal and vertical coordinate distances. The standard radius is reported as a nonnegative number.
4. Why does this converter use atan2?
atan2 uses both y and x values. It determines the correct direction automatically. A simple inverse tangent can return the right reference angle but the wrong quadrant.
5. What happens at the origin?
At (0, 0), the radius is zero. The angle is undefined because every direction begins at the origin. The calculator labels this special case clearly.
6. Which angle range should I choose?
Choose the positive range for angles from 0° up to 360°. Choose the signed range for angles from -180° to 180°. Both can describe the same point.
7. Can polar coordinates use a negative radius?
Some advanced systems allow it. A negative radius reverses the direction by 180 degrees. This page uses the standard positive radius for clear, consistent conversion results.
8. Should I use degrees or radians?
Use degrees for most school geometry and drawings. Use radians for trigonometry, calculus, programming, and many scientific formulas. The converter displays both after calculation.
9. Does the result identify the quadrant?
Yes. The result identifies the quadrant or axis location. This is useful for checking whether the reported angle agrees with the signs of x and y.
10. Are points on axes in a quadrant?
No. A point on an axis does not belong to a quadrant. It has a standard angle such as 0°, 90°, 180°, or 270°.
11. How can I check my conversion?
Substitute radius and angle into x = r cos θ and y = r sin θ. The recovered values should match the originals after rounding. Polar coordinates make directional position checks clear and reliable.