Rectangular to Polar Coordinates Calculator

How to use the rectangular to polar coordinates calculator

What this calculator does

This calculator converts one two-dimensional rectangular, or Cartesian, point (x, y) into the polar form (r, θ). The radius r is the straight-line distance from the origin to the point. The principal angle θ is the counterclockwise direction from the positive x-axis, reported in either degrees or radians. The result is an exact coordinate identity apart from display rounding; it does not draw a full polar graph, solve a polar equation, or choose every mathematically equivalent angle.

When to use it

Use the converter when a geometry, trigonometry, physics, engineering, navigation, complex-number, or graphics problem gives a point in x-y form but the next calculation is easier with distance and direction. It is also useful for checking hand calculations, confirming the quadrant of an angle, or preparing a clean calculation record in Excel. OpenStax provides a broader explanation of polar coordinates and rectangular-to-polar conversion formulas.

How to calculate

1. The calculator opens with the demonstration point (3, 4). Its finite results and example Excel workbook are ready immediately.
2. Replace x coordinate with the horizontal coordinate and y coordinate with the vertical coordinate. Results update live.
3. Choose Angle unit: Degrees for an angle from – 180° through 180°, or Radians for the equivalent principal value from – π through π.
4. Read Radius r, Principal angle θ, and the combined Polar pair (r, θ). Review the calculation table to see the squared components, their sum, the square root, and the quadrant-aware angle operation.
5. Select Download Excel to export the current validated inputs and outputs as a real .xlsx workbook. Select Reset to clear the demonstration values and all calculated content. Reset may disable the Excel button until both required coordinates contain complete valid numbers again.

Input guide

x coordinate is required. Enter a finite real number in coordinate units, such as 3, – 2.5, or 1,250.75. A period is the decimal separator; commas are accepted only as standard three-digit grouping. Increasing x generally rotates the point toward the positive x-axis and may increase the radius. Do not enter scientific notation, unit letters, or an ambiguous decimal comma such as 1,5.

y coordinate is required and follows the same number format, range, and grouping rules. A value such as 4 places the demonstration point above the x-axis. Positive y values point upward; negative y values point downward. Changing y affects both radius and angle. A common error is to ignore the sign, which can place the direction in the wrong quadrant.

Angle unit is a required display control with two valid options: Degrees and Radians. It does not alter x, y, or r. It changes the presentation and workbook value for θ. For example, 53.130102° and approximately 0.927295 radians represent the same direction. The calculator uses the two-argument arctangent rather than plain arctan(y/x); the Math.atan2 definition and quadrant behavior explain why this avoids division-by-zero and quadrant ambiguity.

Output guide

Radius r is the nonnegative distance from the origin, measured in the same coordinate units as x and y. It is driven by both inputs through √(x² + y²). A radius of zero means the point is at the origin; larger absolute coordinate values generally produce a larger radius. Principal angle θ is the direction returned by atan2(y, x). Its sign indicates counterclockwise or clockwise direction within the selected principal range. At the origin, this calculator uses 0 as a practical display convention, while every angle describes the same zero-radius point.

Polar pair (r, θ) combines the two outputs in standard order. The header pills summarize the current rectangular point and selected angle unit. In the breakdown table, x squared and y squared are always nonnegative, while Sum of squares is their total. These three rows support the radius identity. The Radius r row shows the square root, and the Principal angle θ row shows atan2 with y first and x second. The National Institute of Standards and Technology's reference on inverse trigonometric functions and principal values gives additional mathematical context for principal-angle conventions.

Worked example

For the startup example x = 3 and y = 4, the squared components are 3² = 9 and 4² = 16. Their sum is 25, so r = √25 = 5. The direction is θ = atan2(4, 3) = 53.130102°, or approximately 0.927295 radians. The first-open polar coordinate is therefore (5, 53.130102°). The displayed values, table rows, live summary, and Excel workbook all come from the same canonical calculation.