Cartesian to Polar Converter

Cartesian vs Polar Coordinate Systems

The Cartesian coordinate system identifies points on a two-dimensional plane using horizontal and vertical distances $(x, y)$ from a central origin $(0, 0)$. In contrast, the Polar coordinate system identifies points using a straight-line distance $(r)$ from the origin and an angle $(\theta)$ measured counterclockwise from the positive horizontal axis.

Conversion Formulas

To convert from Cartesian $(x, y)$ to Polar $(r, \theta)$, we use the Pythagorean theorem and trigonometry:

1. Calculating the Radius ($r$)

The radial distance is the hypotenuse of a right triangle formed by the $x$ and $y$ coordinates:

$$r = \sqrt{x^2 + y^2}$$

2. Calculating the Angle ($\theta$)

The angle is computed using the multi-valued arctangent function to correctly determine the quadrant based on the signs of $x$ and $y$:

$$\theta = \operatorname{atan2}(y, x)$$

You can explore the detailed behavior of quadrant-aware arctangents using our Arctan2 Calculator.

Quadrant and Axis Conventions

Because simple $\tan^{-1}(y/x)$ only returns values between $-90^\circ$ and $90^\circ$, it does not distinguish between opposite quadrants (e.g. $(3, 4)$ in Quadrant I versus $(-3, -4)$ in Quadrant III). The $\operatorname{atan2}$ function correctly maps angles through the full $360^\circ$ ($2\pi$ radians) range by checking the signs of both inputs:

Quadrant I: $x > 0, y > 0 \implies \theta = \tan^{-1}(y/x)$
Quadrant II: $x < 0, y > 0 \implies \theta = \tan^{-1}(y/x) + 180^\circ$
Quadrant III: $x < 0, y < 0 \implies \theta = \tan^{-1}(y/x) + 180^\circ$
Quadrant IV: $x > 0, y < 0 \implies \theta = \tan^{-1}(y/x) + 360^\circ$

Frequently Asked Questions

What is the difference between atan and atan2?

The standard atan(y/x) function only takes a single ratio, losing the individual signs of x and y, and yields results in only two quadrants (-90° to 90°). Atan2(y, x) takes both coordinates separately to determine the correct direction across all four quadrants.

Can the radius (r) be negative in polar coordinates?

In standard geometric representations, the radius r represents a distance and is always non-negative (r ≥ 0). However, in advanced calculus, negative r is sometimes used to denote a point in the opposite direction (rotated by 180°).

How do I convert polar coordinates back to Cartesian?

You can easily convert polar coordinates back to Cartesian using standard sine and cosine projections: x = r * cos(θ) and y = r * sin(θ).

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