Polar to Cartesian Converter

What is Polar to Cartesian Conversion?

In coordinate geometry, a point can be described in different coordinate systems depending on the application. The two most widely used systems in two dimensions are:

Polar Coordinates $(r, \theta)$: Defined by a radial distance $r$ from the origin and an angular direction $\theta$ measured counter-clockwise from the positive x-axis.
Cartesian Coordinates $(x, y)$: Defined by horizontal $x$ and vertical $y$ displacements on a rectangular grid.

Polar to Cartesian Formulas

Converting polar coordinates $(r, \theta)$ to Cartesian coordinates $(x, y)$ is computed using right-triangle trigonometry:

$$x = r \cdot \cos(\theta)$$

$$y = r \cdot \sin(\theta)$$

Where:

$r$ is the radial distance from the origin.
$\theta$ is the angle with respect to the positive x-axis (measured in degrees or radians).
$x$ is the horizontal rectangular coordinate.
$y$ is the vertical rectangular coordinate.

Extension to 3D Coordinate Systems

1. Cylindrical to Cartesian

Cylindrical coordinates extend 2D polar coordinates by adding a vertical elevation $z$:

$$x = r \cos(\theta), \quad y = r \sin(\theta), \quad z = z$$

2. Spherical to Cartesian

Spherical coordinates use radial distance $\rho$, azimuthal angle $\theta$, and polar/inclination angle $\phi$ measured from the positive z-axis:

$$x = \rho \sin(\phi) \cos(\theta)$$

$$y = \rho \sin(\phi) \sin(\theta)$$

$$z = \rho \cos(\phi)$$

Related Coordinate and Vector Tools

Check out related mathematical and geometry calculators:

Cartesian to Polar Converter: Convert $(x, y)$ coordinates back into $(r, \theta)$.
Cylindrical Coordinates Calculator: Convert between Cartesian, Cylindrical, and Spherical 3D systems.
Angle Between Two Vectors Calculator: Calculate angles and dot products in 2D and 3D space.

Frequently Asked Questions

Can radius (r) be negative in polar coordinates?

Yes. A negative radius $-r$ represents a point located a distance $|r|$ from the origin in the opposite direction, equivalent to $(|r|, \theta + 180^\circ)$ or $(|r|, \theta + \pi)$. The conversion formulas $x = r \cos\theta$ and $y = r \sin\theta$ handle negative radius values accurately without modification.

How do I convert angle degrees to radians?

Multiply degrees by $\pi / 180$: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$. For instance, $90^\circ = \frac{\pi}{2} \approx 1.5708 \text{ rad}$, and $180^\circ = \pi \approx 3.1416 \text{ rad}$.

What happens when the angle is a special angle like 90 degrees?

At $\theta = 90^\circ$ ($\pi/2$ radians), $\cos(90^\circ) = 0$ and $\sin(90^\circ) = 1$, which gives $(x, y) = (0, r)$, placing the point directly on the positive y-axis.

Why are polar coordinates used in physics and engineering?

Polar coordinates simplify calculations involving circular motion, central forces (like gravity or electrostatic orbits), radar navigation, antenna radiation patterns, and fluid vortices where radial symmetry exists.

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