Polar Coordinates Calculator

What is Polar Coordinates Calculator

A Polar Coordinates Calculator is a tool that converts points and equations between polar coordinates \((r,\theta)\) and Cartesian coordinates \((x,y)\), computes radius and angle from a given point, and evaluates polar equations for plotting or analysis. Polar coordinates represent positions using a distance from the origin (radius \(r\)) and an angle \(\theta\) measured from a reference direction (usually the positive x-axis). This calculator helps students, engineers, and scientists quickly transform coordinates, interpret angular relationships, and prepare data for graphs or simulations.

About the Polar Coordinates Calculator

The calculator supports common tasks: converting a single point from polar to Cartesian or vice versa, converting entire polar equations into parametric Cartesian form, evaluating \(r(\theta)\) at sampled angles for plotting, and normalizing angles to preferred ranges (for example \([0,2\pi)\) or \((-\pi,\pi]\)). It handles degrees or radians, resolves angle quadrant ambiguities using two-argument arctangent, and displays results in both exact symbolic form and decimal approximations. Formulas are provided in render-ready math markup so they display clearly and responsively with a math renderer.

How to Use this Polar Coordinates Calculator

1. Choose conversion direction: Polar → Cartesian or Cartesian → Polar.
2. Enter the point coordinates (for polar provide \(r\) and \(\theta\); for Cartesian provide \(x\) and \(y\)). Select degrees or radians.
3. Click Calculate. For polar-to-Cartesian the calculator returns \(x\) and \(y\); for Cartesian-to-polar it returns \(r\) (nonnegative) and \(\theta\) adjusted to the chosen range. For polar equations you may input \(r(\theta)\) and request a parametric conversion \(x(\theta),y(\theta)\) or a sampled table.
4. Use results for plotting, physics problems involving radial components, converting velocity/force between coordinate systems, or preparing datasets.

Formulas (render-ready)

Polar → Cartesian:
\( \; x = r\cos\theta,\quad y = r\sin\theta \; \)

Cartesian → Polar:
\( \; r = \sqrt{x^2 + y^2},\quad \theta = \operatorname{atan2}(y,x) \; \)

Parametric form for a polar equation \( r(\theta) \):
\( \; x(\theta) = r(\theta)\cos\theta,\quad y(\theta) = r(\theta)\sin\theta \; \)

Examples of Polar Coordinates Calculator

Example 1 — Polar to Cartesian:
Given \( (r,\theta) = \bigl(5,\tfrac{\pi}{6}\bigr) \): \( x = 5\cos(\tfrac{\pi}{6}) = \dfrac{5\sqrt{3}}{2} \approx 4.3301,\; y = 5\sin(\tfrac{\pi}{6}) = 2.5 \).

Example 2 — Cartesian to Polar:
Given \( (x,y)=(-3,3\sqrt{3}) \): \( r=\sqrt{(-3)^2+(3\sqrt{3})^2}=\sqrt{9+27}=6 \). Angle \( \theta=\operatorname{atan2}(3\sqrt{3},-3)=\tfrac{2\pi}{3} \) (120°).

Example 3 — Polar equation sampled and converted:
For \( r(\theta)=2+3\cos\theta \), parametric form is \( x(\theta)=(2+3\cos\theta)\cos\theta,\; y(\theta)=(2+3\cos\theta)\sin\theta \). Sample \(\theta\) to plot the curve.

Responsive formula notes

The formulas above are provided in math markup that will scale and reflow when rendered by a math renderer on the page. Keep equations in inline or display math mode so they remain legible on phones, tablets, and desktops.

What is the difference between polar and Cartesian coordinates?

Cartesian uses \((x,y)\) positions; polar uses distance \(r\) from origin and angle \(\theta\). Either can represent the same point via conversion formulas.

How do I convert polar to Cartesian?

Use \(x=r\cos\theta\) and \(y=r\sin\theta\). Ensure \(\theta\) is in the correct units (degrees or radians) for trigonometric functions.

How do I convert Cartesian to polar?

Compute \(r=\sqrt{x^2+y^2}\) and \(\theta=\operatorname{atan2}(y,x)\) to get the correct quadrant for the angle.

What is atan2 and why use it?

\(\operatorname{atan2}(y,x)\) returns an angle accounting for the signs of \(x\) and \(y\), avoiding quadrant ambiguity present in simple arctangent.

Can r be negative in polar coordinates?

Some conventions allow negative \(r\) (which flips angle by \(\pi\)); most calculators return \(r\ge0\) and adjust \(\theta\) accordingly for consistency.

How do I convert a polar equation to Cartesian form?

Use \(x=r\cos\theta\) and \(y=r\sin\theta\); substitute \(r=\sqrt{x^2+y^2}\) where needed to eliminate polar variables and obtain Cartesian relations.

Are polar coordinates useful in physics?

Yes — they simplify problems with radial symmetry (circular motion, central forces, fields around a point) by separating radial and angular components.