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.

Your Input

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 $$$\frac{\pi}{3} + 2 \pi = \frac{7 \pi}{3}$$$.

Answer

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

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