This all-in-one online Polar Coordinates Calculator converts polar to Cartesian coordinates and Cartesian to polar coordinates.
Polar System of Coordinates
The polar coordinate system is a two-dimensional coordinate system in which each point \(P\) on a plane is determined by a distance \(r\) from a reference point and an angle \(\theta\) from a reference direction.
The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are commonly expressed in either degrees or radians.
In mathematics, the reference direction is usually drawn as a ray from the pole horizontally to the right (along the x-axis), and the polar angle \(\theta\) increases for counterclockwise rotations. The polar angle decreases towards negative values for rotations in the opposite direction.
On the other hand, as we know, rectangular or Cartesian coordinates on a plane are a pair of numbers \(x\) and \(y\) that are measured along the x-axis and y-axis respectively. These two axes are perpendicular to each other. Each point \(P\) on a plane can be determined by these pair of coordinates.
The polar coordinates \(r\) and \(\theta\) can be converted to the Cartesian coordinates \(x\) and \(y\) by using the trigonometric functions sine and cosine:
$$x = r \cdot cos\theta, \ y = r \cdot sin\theta.$$
The Cartesian coordinates \(x\) and \(y\) can be converted to polar coordinates \(r\) and \(\theta\) with \(r\) ≥ 0 and \(\theta\) in the interval (−\(\pi\), \(\pi\)] using the following formulas:
$$r = \sqrt{x^{2}+y^{2}},$$
$$\theta = arctan(\frac{y}{x}).$$
If \(r\) is calculated first as above, then the formula for \(\theta\) can be expressed more simply using the arccosine function:
$$\theta = \left\{\begin{matrix}
arccos(\frac{x}{r}), \ if \ y \geq 0 \ and \ r\neq 0,\\
\\
-arccos(\frac{x}{r}), \ if \ y < 0, \qquad \qquad \\
\\
undefined, \ if \ r = 0. \qquad \qquad \\
\end{matrix}\right.$$
These formulas are used in our Polar Coordinates Calculator, which makes coordinate transformations a matter of fractions of a second.
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