Polar/Rectangular Coordinates Calculator

Convert polar coordinates to/from rectangular step by step

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

Related calculator: Polar/Rectangular Equation Calculator

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Your Input

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

Solution

We have that ρ=x2+y2=12+(3)2=2\rho = \sqrt{x^{2} + y^{2}} = \sqrt{1^{2} + \left(\sqrt{3}\right)^{2}} = 2.

Next, θ=atan(yx)=atan(31)=π3\theta = \operatorname{atan}{\left(\frac{y}{x} \right)} = \operatorname{atan}{\left(\frac{\sqrt{3}}{1} \right)} = \frac{\pi}{3}.

It is also possible that ρ\rho is negative. In this case, add/subtract π\pi from the found θ\theta: θ=π3+π=4π3\theta = \frac{\pi}{3} + \pi = \frac{4 \pi}{3}.

NOTE: all found angles are in the interval [0,2π)\left[0, 2 \pi\right). If you need angles in another interval, add/subtract 2π2 \pi the required number of times.

For example, π3\frac{\pi}{3} in the interval [2π,4π)\left[2 \pi, 4 \pi\right) is π3+2π=7π3\frac{\pi}{3} + 2 \pi = \frac{7 \pi}{3}.

Answer

(ρ,θ)=(2,π3)(2,1.047197551196598)\left(\rho, \theta\right) = \left(2, \frac{\pi}{3}\right)\approx \left(2, 1.047197551196598\right)A

(ρ,θ)=(2,4π3)(2,4.188790204786391)\left(\rho, \theta\right) = \left(-2, \frac{4 \pi}{3}\right)\approx \left(-2, 4.188790204786391\right)A