Rectangular to Polar Converter

Convert Cartesian (x, y) coordinates to polar (r, θ) with quadrant identification and step-by-step solutions.

Enter Rectangular Coordinates

Polar Coordinates

Polar Form (r, θ)
--
r (radius) --
θ (degrees) --
θ (radians) --
Quadrant --

Step-by-Step Solution

Rectangular to Polar Conversion

In mathematics, the rectangular (Cartesian) coordinate system uses (x, y) to represent points on a plane, while the polar coordinate system uses (r, θ) -- a distance from the origin and an angle from the positive x-axis. Converting between these systems is fundamental in trigonometry, physics, and engineering.

Conversion Formulas

Radius (r)

Distance from the origin to the point.

r = √(x² + y²)

Angle (θ)

Angle from the positive x-axis.

θ = atan2(y, x)

Reverse Conversion

From polar back to rectangular.

x = r cosθ, y = r sinθ

Understanding Quadrants

The angle θ depends on which quadrant the point falls in:

  • Quadrant I (x > 0, y > 0): θ is between 0° and 90°.
  • Quadrant II (x < 0, y > 0): θ is between 90° and 180°.
  • Quadrant III (x < 0, y < 0): θ is between -180° and -90° (or 180° to 270°).
  • Quadrant IV (x > 0, y < 0): θ is between -90° and 0° (or 270° to 360°).

Why atan2 Instead of atan?

The standard arctangent function (atan) only returns values between -90° and 90°, making it impossible to distinguish between Quadrants I/III or II/IV. The atan2(y, x) function uses the signs of both x and y to determine the correct quadrant, returning the full range of angles from -180° to 180°.

Applications

  • Physics: Describing circular and rotational motion.
  • Navigation: Bearing and distance calculations.
  • Signal Processing: Representing complex numbers in polar form.
  • Computer Graphics: Rotation and transformation of objects.