How to convert polar equations to rectangular equations

In polar coordinates, a point is given by a distance and an angle. We’ll use the connection between those coordinates and x and y to rewrite an equation without changing the curve it describes.

Start with the three conversion formulas

The formulas connect the polar coordinates $r$ and $θ$ to the rectangular coordinates $x$ and $y$ . Look for a part of the equation that already matches one of them, or multiply by $r$ to make a match.

\begin{array}{l} x = r \cos (θ) \\ y = r \sin (θ) \\ r^{2} = x^{2} + y^{2} \end{array}

Convert the polar equation of a circle

Start with $r = 3 \cos (θ)$ . Multiplying both sides by $r$ gives us $r \cos (θ)$ , which we can replace with $x$ . We can then replace $r^{2}$ with $x^{2} + y^{2}$ .

\begin{array}{l} r^{2} = 3 r \cos (θ) = 3 x \\ x^{2} + y^{2} = 3 x \end{array}

The origin satisfies the new equation, and it is on the original polar curve because $r = 0$ when $θ = π / 2$ . Completing the square makes the shape easier to recognize.

\begin{array}{l} {(x - \frac{3}{2})}^{2} + y^{2} = \frac{9}{4} \end{array}

The rectangular equation describes the same circle as the polar equation.

Use the double-angle identity when you see twice the angle

For $r^{2} = \sin (2 θ)$ , use $\sin (2 θ) = 2 \sin (θ) \cos (θ)$ . Multiplying by $r^{2}$ gives us a factor of $r$ to pair with each trigonometric function.

\begin{array}{l} r^{4} = 2 (r \sin (θ)) (r \cos (θ)) \\ {(x^{2} + y^{2})}^{2} = 2 x y \end{array}

The origin is on both curves. For every other point, $r^{2} > 0$ , so we can also divide by $r^{2}$ to go back to the polar equation.

To work in the opposite direction, see how to convert rectangular equations to polar equations.

Learn mathKrista KingJune 24, 2021math, learn online, online course, online math, calculus 2, calculus ii, calc 2, calc ii, polar and parametric, polar curves, polar equations, converting polar equations, polar equations into rectangular equations, polar to rectangular, converting polar to rectangular, converting polar to cartesian, polar to cartesian

How to convert polar equations to rectangular equations

Start with the three conversion formulas

Convert the polar equation of a circle

Use the double-angle identity when you see twice the angle

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