Convert between Cartesian and polar coordinates
Last updated: March 2026
Cartesian Coordinates (x, y):
(3, 4)
Polar Coordinates (r, θ):
r = 5
θ = 53.130102° = 0.927295 rad
Conversion Formulas:
r = √(x² + y²) = 5
θ = atan2(y, x) = 0.927295 rad
Polar coordinates represent a point on a plane using two values: a distance from the origin called the radius r, and an angle θ measured from the positive x-axis. This is an alternative to Cartesian coordinates (x, y).
Polar coordinates are especially useful for circular motion, rotational symmetry, and trigonometric modeling. Conversions use x = r·cos(θ), y = r·sin(θ), r = √(x² + y²), and θ = atan2(y, x).
Select your input coordinate system
Choose between Cartesian (x, y) mode or Polar (r, θ) mode at the top of the calculator. This determines which values you'll enter first.
Why: Different problems start with different coordinate types. Cartesian is intuitive for rectangular grids, while polar is natural for circular or rotational problems.
Enter your known coordinates
Input the x and y values (for Cartesian) or radius and angle values (for Polar). The calculator accepts decimal and negative numbers as appropriate.
Why: You need starting values to perform the coordinate transformation. The accuracy of your input directly affects the conversion result.
Select the angle unit (for polar input)
If using Polar mode, choose whether your angle is in degrees (°) or radians (rad). This setting only applies when entering polar coordinates.
Why: Different fields use different angle units—degrees are common in navigation and design, while radians are standard in mathematics and physics.
View the converted coordinates and formulas
The calculator instantly displays the converted values in the other coordinate system along with the mathematical formulas used (r = √(x² + y²), θ = atan2(y, x), x = r·cos(θ), y = r·sin(θ)).
Why: Seeing both the results and the underlying formulas reinforces your understanding and helps you verify calculations manually if needed.
Apply conversions to your problem
Use the converted coordinates and formulas for your specific application—whether plotting a point, rotating in graphics, analyzing circular motion, or solving trigonometric problems.
Why: Mastering coordinate conversion makes you fluent in both systems, allowing you to choose the most efficient approach for any geometric or physics problem.
An air traffic controller is monitoring aircraft on a radar system centered at the airport. An aircraft is detected at 50 km distance (radius) and at an angle of 45° from the positive x-axis (east direction). The controller needs to convert this radar position (polar coordinates) to Cartesian coordinates to plot it accurately on a standard navigation map and communicate the position to other systems.
Step 1 — Record radar measurements:
Radar provides distance r = 50 km and angle θ = 45° measured counterclockwise from east.
Step 2 — Apply x-coordinate conversion formula:
x = r·cos(θ) = 50·cos(45°) = 50·(0.707107) ≈ 35.355 km
Step 3 — Apply y-coordinate conversion formula:
y = r·sin(θ) = 50·sin(45°) = 50·(0.707107) ≈ 35.355 km
Step 4 — Plot Cartesian coordinates on map:
Point: (35.355, 35.355) kilometers northeast of airport
Step 5 — Communicate position in multiple formats:
Controller reports: "Aircraft at 50 km bearing 45°" (radar format) AND "Aircraft at position (35.355, 35.355)" (map format for other systems).
Verify by converting back: r = √(35.355² + 35.355²) = √(1250 + 1250) = √2500 = 50 km ✓. θ = atan2(35.355, 35.355) = 45° ✓. Coordinates are verified.
Converted coordinates: (35.355 km, 35.355 km)
The aircraft is located 35.355 km east and 35.355 km north of the airport. Since both components are equal at 45°, the aircraft is along the northeast diagonal.
The conversion reveals that at 45°, the aircraft has equal components in both horizontal and vertical directions. The Cartesian representation allows the controller to estimate minimal distances to other aircraft, calculate approach vectors, and integrate with navigation systems using rectangular coordinates. On a standard map, this aircraft appears in the northeast quadrant with easily calculated distances to runways, checkpoints, and other aircraft positions. The controller can now confidently direct this aircraft and coordinate with other air traffic management systems using the universally understood Cartesian format.
Cartesian coordinates use x and y distances, while polar coordinates use radius and angle. Both describe the same point.
This calculator uses non-negative radius values, which is the standard convention for most applications.
atan2(y, x) returns the correct angle for all four quadrants.
Yes, in standard mathematics. Some navigation systems use different reference directions.
Use whichever unit matches your problem. The calculator supports both.
Angles can wrap around, but standard ranges are 0° to 360° or 0 to 2π radians.
Use polar coordinates for circular motion, rotations, and radial geometry problems.
Start at the origin, rotate by θ, and move outward by distance r.
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