Polar/Rectangular Coordinates Calculator

The calculator will convert the polar coordinates to rectangular (Cartesian) and vice versa, with steps shown.

Related calculator: Polar/Rectangular Equation Calculator

(
,
)

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Convert $\left(x, y\right) = \left(1, \sqrt{3}\right)$ to polar coordinates.

Solution

We have that $\rho = \sqrt{x^{2} + y^{2}} = \sqrt{1^{2} + \left(\sqrt{3}\right)^{2}} = 2$.

Next, $\theta = \operatorname{atan}{\left(\frac{y}{x} \right)} = \operatorname{atan}{\left(\frac{\sqrt{3}}{1} \right)} = \frac{\pi}{3}$.

It is also possible that $\rho$ is negative. In this case, add/subtract $\pi$ from the found $\theta$: $\theta = \frac{\pi}{3} + \pi = \frac{4 \pi}{3}$.

NOTE: all found angles are in the interval $\left[0, 2 \pi\right)$. If you need angles in another interval, add/subtract $2 \pi$ the required number of times.

For example, $\frac{\pi}{3}$ in the interval $\left[2 \pi, 4 \pi\right)$ is $\frac{\pi}{3} + 2 \pi = \frac{7 \pi}{3}$.

$\left(\rho, \theta\right) = \left(2, \frac{\pi}{3}\right)\approx \left(2, 1.047197551196598\right)$A
$\left(\rho, \theta\right) = \left(-2, \frac{4 \pi}{3}\right)\approx \left(-2, 4.188790204786391\right)$A