How to Convert a Rectangular Equation to Polar Form (and Back) with 16 Worked Examples

Quick answer: To convert a rectangular equation to polar, replace every $x$ with $r\cos\theta$, every $y$ with $r\sin\theta$, and every $x^2 + y^2$ with $r^2$, then solve for $r$. To convert a polar equation to rectangular, multiply or rearrange until you can replace $r\cos\theta$ with $x$, $r\sin\theta$ with $y$, and $r^2$ with $x^2 + y^2$.

Example: $x^2 + y^2 = 36$ becomes $r^2 = 36$, so the polar equation is $r = 6$.

Converting a single point such as $(3, 4)$ is a calculation, and our rectangular to polar calculator does it instantly. Converting an equation such as $y = x^2$ is algebra: you are rewriting the description of an entire curve. This guide covers that algebra with 16 worked examples, eight in each direction.

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The Conversion Rules

Every conversion uses the same small set of identities. They come from the right triangle formed by a point, the origin, and the x-axis (see rectangular to polar formulas for the derivation).

Rectangular	Polar	Use it when you see…
$x$	$r\cos\theta$	a lone $x$
$y$	$r\sin\theta$	a lone $y$
$x^2 + y^2$	$r^2$	a sum of squares (circles)
$\dfrac{y}{x}$	$\tan\theta$	a ratio, or a line through the origin

Going the other way, you look for these same groupings in the polar equation: $r\cos\theta \to x$, $r\sin\theta \to y$, $r^2 \to x^2 + y^2$, and $r = \sqrt{x^2 + y^2}$.

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Part 1: Rectangular Equation to Polar Equation

Method

1. Substitute $x = r\cos\theta$ and $y = r\sin\theta$. Replace $x^2 + y^2$ with $r^2$ wherever it appears.
2. Simplify, using identities such as $\cos^2\theta + \sin^2\theta = 1$.
3. Solve for $r$ if possible. If you divide by $r$, check whether the pole ($r = 0$) still lies on the curve.

Example 1: Circle centered at the origin, $x^2 + y^2 = 36$

$$x^2 + y^2 = 36 \;\Rightarrow\; r^2 = 36 \;\Rightarrow\; r = 6$$

A circle of radius 6 around the origin is simply $r = 6$. The same method gives $x^2 + y^2 = 64 \Rightarrow r = 8$ and $x^2 + y^2 = 25 \Rightarrow r = 5$. We use $r = 6$ rather than $r = \pm 6$ because $r = -6$ traces the same circle.

Example 2: Horizontal line, $y = 5$

$$r\sin\theta = 5 \;\Rightarrow\; r = \frac{5}{\sin\theta} = 5\csc\theta$$

Example 3: Vertical line, $x = 3$

$$r\cos\theta = 3 \;\Rightarrow\; r = 3\sec\theta$$

Example 4: Line through the origin, $y = x$

$$r\sin\theta = r\cos\theta \;\Rightarrow\; \tan\theta = 1 \;\Rightarrow\; \theta = \frac{\pi}{4}$$

A line through the origin has no $r$ in its polar equation. Every point on it has the same direction. Negative values of $r$ cover the half of the line in Quadrant III.

Example 5: Line not through the origin, $y = x + 1$

$$r\sin\theta = r\cos\theta + 1 \;\Rightarrow\; r(\sin\theta - \cos\theta) = 1 \;\Rightarrow\; r = \frac{1}{\sin\theta - \cos\theta}$$

In general, the line $ax + by = c$ becomes $r = \dfrac{c}{a\cos\theta + b\sin\theta}$.

Example 6: Circle through the origin, $x^2 + y^2 = 4x$

$$r^2 = 4r\cos\theta \;\Rightarrow\; r = 4\cos\theta$$

We divided by $r$, which could lose the solution $r = 0$. But $r = 4\cos\theta$ reaches $r = 0$ at $\theta = \frac{\pi}{2}$, so the pole is still included. This is a circle of radius 2 centered at $(2, 0)$.

Example 7: Circle through the origin, $x^2 + y^2 = 6y$

$$r^2 = 6r\sin\theta \;\Rightarrow\; r = 6\sin\theta$$

This is a circle of radius 3 centered at $(0, 3)$. More generally, $x^2 + y^2 + Dx + Ey = 0$ becomes $r = -D\cos\theta - E\sin\theta$.

Example 8: Parabola, $y = x^2$

$$r\sin\theta = r^2\cos^2\theta \;\Rightarrow\; r = \frac{\sin\theta}{\cos^2\theta} = \tan\theta\,\sec\theta$$

Bonus: Hyperbolas $xy = 4$ and $x^2 - y^2 = 1$

$$xy = 4 \;\Rightarrow\; r^2\cos\theta\sin\theta = 4 \;\Rightarrow\; \frac{r^2}{2}\sin 2\theta = 4 \;\Rightarrow\; r^2 = 8\csc 2\theta$$ $$x^2 - y^2 = 1 \;\Rightarrow\; r^2(\cos^2\theta - \sin^2\theta) = 1 \;\Rightarrow\; r^2\cos 2\theta = 1 \;\Rightarrow\; r^2 = \sec 2\theta$$

Both use double-angle identities: $2\sin\theta\cos\theta = \sin 2\theta$ and $\cos^2\theta - \sin^2\theta = \cos 2\theta$.

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Part 2: Polar Equation to Rectangular Equation

Method

1. Look for $r\cos\theta$, $r\sin\theta$ or $r^2$ already in the equation.
2. If they are not there, multiply both sides by $r$ (or rearrange) to create them.
3. Replace them with $x$, $y$ and $x^2 + y^2$, then simplify. Complete the square to identify circles.

Example 9: $r = 4$

$$r^2 = 16 \;\Rightarrow\; x^2 + y^2 = 16$$

A circle of radius 4 centered at the origin.

Example 10: $\theta = \dfrac{\pi}{3}$

$$\tan\theta = \tan\frac{\pi}{3} = \sqrt{3} \;\Rightarrow\; \frac{y}{x} = \sqrt{3} \;\Rightarrow\; y = \sqrt{3}\,x$$

Example 11: $r = 2\sec\theta$

$$r = \frac{2}{\cos\theta} \;\Rightarrow\; r\cos\theta = 2 \;\Rightarrow\; x = 2$$

Example 12: $r = 4\sin\theta$

Multiply both sides by $r$:

$$r^2 = 4r\sin\theta \;\Rightarrow\; x^2 + y^2 = 4y \;\Rightarrow\; x^2 + (y - 2)^2 = 4$$

A circle of radius 2 centered at $(0, 2)$.

Example 13: $r = 6\cos\theta$

$$r^2 = 6r\cos\theta \;\Rightarrow\; x^2 + y^2 = 6x \;\Rightarrow\; (x - 3)^2 + y^2 = 9$$

A circle of radius 3 centered at $(3, 0)$.

Example 14: $r = \dfrac{2}{\sin\theta + \cos\theta}$

$$r\sin\theta + r\cos\theta = 2 \;\Rightarrow\; y + x = 2$$

A straight line with slope $-1$ and y-intercept 2.

Example 15: $r = \dfrac{1}{1 - \cos\theta}$

$$r - r\cos\theta = 1 \;\Rightarrow\; r = 1 + x \;\Rightarrow\; x^2 + y^2 = (1 + x)^2 \;\Rightarrow\; y^2 = 1 + 2x$$

A parabola opening to the right. Squaring both sides of $r = 1 + x$ is safe here because $r \ge 0$ and $1 + x = r \ge 0$ on this curve.

Example 16: $r^2 = 4\sin 2\theta$ (lemniscate)

$$r^2 = 8\sin\theta\cos\theta \;\Rightarrow\; r^4 = 8(r\sin\theta)(r\cos\theta) \;\Rightarrow\; (x^2 + y^2)^2 = 8xy$$

We multiplied both sides by $r^2$ so that each $\sin\theta$ and $\cos\theta$ gets its own factor of $r$.

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Summary Table

Rectangular equation	Polar equation	Shape
$x^2 + y^2 = a^2$	$r = a$	Circle at origin
$x = a$	$r = a\sec\theta$	Vertical line
$y = b$	$r = b\csc\theta$	Horizontal line
$y = mx$	$\theta = \arctan m$	Line through origin
$x^2 + y^2 = 2ax$	$r = 2a\cos\theta$	Circle through origin, center $(a, 0)$
$x^2 + y^2 = 2by$	$r = 2b\sin\theta$	Circle through origin, center $(0, b)$
$y = x^2$	$r = \tan\theta\sec\theta$	Parabola
$x^2 - y^2 = 1$	$r^2 = \sec 2\theta$	Hyperbola

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Common Mistakes

1. Substituting $r$ for $x^2 + y^2$. The correct replacement is $r^2$. Only $\sqrt{x^2 + y^2}$ equals $r$.
2. Dividing by $r$ without checking the pole. In $r^2 = 4r\cos\theta$, dividing by $r$ removes the solution $r = 0$. It is safe only if the simplified equation still passes through the pole, as $r = 4\cos\theta$ does at $\theta = \frac{\pi}{2}$.
3. Writing $\theta = \arctan(y/x)$ for every point. Inside an equation, $\tan\theta = y/x$ is fine. When you need an actual angle for a specific point, $\arctan$ alone gives the wrong quadrant for $x < 0$. Use atan2.
4. Forgetting to complete the square. $x^2 + y^2 = 4y$ is correct but hides the shape. $x^2 + (y-2)^2 = 4$ shows the center and radius.
5. Squaring without checking signs. Squaring both sides can add points that are not on the original curve. Check the sign condition, as in Example 15.

More pitfalls that apply to points as well as equations are collected in common mistakes in rectangular and polar conversions.

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Check Your Work with Points

A quick way to test a converted equation is to pick a point on the original curve, convert it, and confirm it satisfies the new equation.

For Example 6, the point $(4, 0)$ lies on $x^2 + y^2 = 4x$ because $16 + 0 = 16$. Converting it with our online calculator gives $r = 4,\ \theta = 0^\circ$. Plugging into $r = 4\cos\theta$ gives $4\cos 0 = 4$, so the conversion checks out.

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Frequently Asked Questions

What is the polar form of $x^2 + y^2 = 36$?

$r = 6$. Replace $x^2 + y^2$ with $r^2$ to get $r^2 = 36$, then take the positive root.

How do you convert $y = x$ to polar form?

Substitute to get $r\sin\theta = r\cos\theta$, so $\tan\theta = 1$ and $\theta = \frac{\pi}{4}$ (45°).

How do you convert $r = 4\sin\theta$ to rectangular form?

Multiply both sides by $r$ to get $r^2 = 4r\sin\theta$, then substitute to get $x^2 + y^2 = 4y$, which is the circle $x^2 + (y - 2)^2 = 4$.

Is there a calculator for converting equations?

Our rectangular to polar calculator converts individual points and shows every step. Equations need the algebra above, but you can use the calculator to check any point on your curve. For the reverse direction with points, see polar to rectangular formulas and examples.