Enter rectangular coordinates
Use x for horizontal movement and y for vertical movement. Results appear above this form after conversion.
How rectangular values become polar values
r = √(x² + y²)
The radius is the straight-line distance from the origin.
θ = atan2(y, x)
This method uses both signs to place the angle correctly.
For a north-based bearing, use bearing = 90° − θ, then normalize the result from 0° to 360°.
Convert a point in four simple steps
- Enter the rectangular x coordinate.
- Enter the rectangular y coordinate.
- Select the angle unit, style, and decimal precision.
- Select Convert to Polar and review the result above.
Common rectangular-to-polar conversions
| Rectangular point | Radius | Positive angle | Location |
|---|---|---|---|
| (3, 4) | 5 | 53.130° | Quadrant I |
| (-3, 4) | 5 | 126.870° | Quadrant II |
| (-4, -3) | 5 | 216.870° | Quadrant III |
| (0, -6) | 6 | 270° | Negative y-axis |
Converting Rectangular Coordinates to Polar Form
Understanding Rectangular and Polar Coordinates
Rectangular coordinates describe a point through horizontal and vertical movement. The x value shows movement left or right. The y value shows movement up or down. Polar coordinates describe the same point with distance and direction. The distance from the origin is the radius. The direction is an angle from a reference line. Both systems identify one exact location. They simply present that location in different ways. A quick conversion helps when a problem changes coordinate systems.
Why This Conversion Matters
Many technical tasks use polar values because direction is important. Engineers use them for forces and rotating parts. Surveyors use them for bearings and locations. Programmers use them for circular paths and animations. Students use them in trigonometry and complex numbers. Rectangular values are often easier to plot. Polar values are often easier to interpret around a center. Converting between them lets you choose the clearest format. The calculator gives consistent results for each coordinate pair.
Reading the Radius and Angle
The radius tells you how far the point lies from the origin. A radius of zero means the point is exactly at the origin. The angle tells you where the point points. Positive angles normally move counterclockwise from the positive x-axis. A signed angle keeps negative directions visible. A normalized angle stays between zero and 360 degrees. Bearing mode measures clockwise from north. Each option can be useful. Degrees are common in practical work. Radians are common in higher mathematics. Select the style that matches your problem statement.
Choosing Accurate Inputs
Enter x and y as ordinary numbers. Decimals are accepted for precise coordinates. Negative values are also accepted. The signs determine the quadrant. A positive x and positive y point lies in quadrant one. A negative x and positive y point lies in quadrant two. A negative x and negative y point lies in quadrant three. A positive x and negative y point lies in quadrant four. The origin has no unique direction. Check units before comparing your answer with another source.
Using Results in Real Problems
Polar results are useful for vectors, navigation, and circular designs. The radius can represent distance, magnitude, or reach. The angle can represent rotation, heading, or orientation. In complex-number work, polar notation reveals multiplication patterns. In physics, force direction becomes easier to read. In graphics, polar motion can create arcs and spirals. The result section shows both a compact polar form and detailed values. It also identifies the quadrant. Use the shown calculation steps to verify your work. Round only when your final task allows it.
Good Checking Habits
Review the coordinate pair before submitting. A reversed sign changes direction. Verify the angle unit matches your assignment. Use more decimal places for technical work. Use fewer for classroom exercises. Compare radius with coordinate sizes. The radius cannot be smaller than either absolute coordinate value. At the origin, report zero radius and an undefined direction. Save results as CSV when you need a record. Print page for a paper or PDF copy.
Rectangular to polar questions
1. What are rectangular coordinates?
Rectangular coordinates use an ordered pair, written as (x, y). The x value gives horizontal position. The y value gives vertical position. They locate a point relative to the origin on two perpendicular axes.
2. What are polar coordinates?
Polar coordinates use a radius and an angle. The radius measures distance from the origin. The angle gives direction. They are written as (r, θ) or in another chosen angle notation.
3. Why is atan2 used instead of arctangent alone?
atan2 uses both x and y signs. It identifies the correct quadrant automatically. Ordinary arctangent can return the same reference angle for points in different quadrants, which may produce a wrong direction.
4. Can the radius be negative?
This calculator reports a nonnegative radius. Polar mathematics can use a negative radius with an angle shifted by 180 degrees. Keeping radius positive makes standard results easier to compare.
5. What happens at the origin?
The radius is zero at the origin. The direction is undefined because every angle starts at the same point. The calculator reports the radius and clearly marks the angle as undefined.
6. What is a normalized angle?
A normalized angle is placed within a selected interval. This calculator uses 0 through 360 degrees for positive angles. It makes equivalent rotations easier to compare and report consistently.
7. When should I choose radians?
Choose radians for calculus, advanced trigonometry, physics formulas, and many programming tasks. Choose degrees for common geometry, bearings, drafting, and classroom problems that state angles in degrees.
8. How is bearing different from a standard angle?
A standard angle starts at the positive x-axis and turns counterclockwise. A bearing starts at north and turns clockwise. Both describe direction, but they use different reference lines and rotation directions.
9. Can I use decimal coordinates?
Yes. Enter integers or decimals for x and y. The calculator accepts negative decimals too. Select more precision when the input measurements or required answer need extra detail.
10. How can I check a polar answer?
Convert back with x = r cos θ and y = r sin θ. The returned x and y should match the original values after normal rounding. Also verify that the displayed quadrant matches the coordinate signs.
11. Does the CSV file include my conversion?
Yes. After a successful conversion, Download CSV creates a simple file containing the input coordinate, polar result, radius, selected angle, principal angle, location, and complex-number form.