## Converting Between Rectangular and Polar Coordinates

17calculus > svc > polar coordinates > converting

### Videos

PatrickJMT videos

-     A03

-     A06    -     B02

B03    -     A07    -

IntegralCALC videos

A10    -     A11    -     A12    -     A13    -     A14

### Search 17calculus

equations

$$x = r \cos(\theta)$$

$$r^2 = x^2 + y^2$$

$$y = r \sin(\theta)$$

$$\theta= \arctan(y/x)$$

coming early 2014 - new mobile site

On this page, we discuss converting between polar and rectangular coordinates in two main areas, converting points and converting equations. Here is a table showing the main topics with links to jump to the topic.

Converting Points In The Plane rectangular → polar

As discussed on the main polar coordinates page, the two main equations we use to convert between polar and rectangular (cartesian) coordinates are

 $$x = r \cos(\theta)$$ $$y = r \sin(\theta)$$

All other equations can be derived from these two equations.

This video clip (less than a minute long) explains really well how to use trigonometry and the Pythagorean Theorem to get these equations.

Rectangular → Polar Points

In this case, we are given a specific point $$(x,y)$$ and we want to find $$r$$ and $$\theta$$ that describes this point. This direction is a little more complicated because we need to pay special attention to the signs of $$x$$ and $$y$$.

The equations we use are $$r = \sqrt{x^2+y^2}$$ and $$\theta= \arctan(y/x)$$. Let's discuss these equations in more detail. This discussion is critical for you to understand in order to correctly determine the polar coordinates.

The first equation looks easy but there is a hidden assumption that you need to be aware of. Looking at the graph on the right, you know from the Pythagorean Theorem that $$r^2 = x^2 + y^2$$. So far, so good. Now, to solve for $$r$$ we take the square root of both sides.

$$\begin{array}{rcl} r^2 & = & x^2 + y^2 \\ \sqrt{r^2} & = & \pm \sqrt{x^2 + y^2} \\ r & = & \pm \sqrt{x^2 + y^2} \end{array}$$

In the second line above, we could have put the $$\pm$$ sign on either side, but to be strictly true, we needed it. (Do you see why?)

Now, in the equation $$r=\sqrt{x^2+y^2}$$ we dropped the $$\pm$$ sign. What does that mean? That means we are assuming that $$r$$ is always positive. So, when we convert from rectangular to polar coordinates, we will take $$r$$ to be positive. This is a subtle point but you need to keep that in mind. (As a teacher, one of my favorite questions on homework or exams will be to ask what happens when $$r$$ is negative. See the practice problems below for examples of this case.)

Okay, now for the equation $$\theta= \arctan(y/x)$$. This takes special care. You cannot just divide $$y$$ by $$x$$ and plug that value into your calculator. You need to know the sign of both $$x$$ and $$y$$, which will determine the quadrant of your answer and thus the angle $$\theta$$. For discussion here, we want the angle to be in the interval $$(-\pi, \pi]$$. Notice this is a half-open (or half-closed) interval and includes $$\pi$$ but excludes $$-\pi$$. Of course, you need to check with your instructor to see what they require.

The best way that we've found to determine this angle, can be found using these equations. [ source ]

$$\theta = \left\{ \begin{array}{lll} \arctan(y/x) & x > 0 & \text{quadrant 1 or 4} \\ \arctan(y/x)+\pi & x < 0, y \geq 0 & \text{quadrant 2} \\ \arctan(y/x)-\pi & x < 0, y < 0 & \text{quadrant 3} \\ +\pi/2 & x = 0, y > 0 & \\ -\pi/2 & x = 0, y < 0 & \\ 0 & x=0, y=0 & \end{array} \right.$$

Rectangular → Polar Equations

In this case, we are given an equation $$f(x,y)=0$$ and we need to find an equation $$g(r, \theta)=0$$. This is pretty easy. We just substitute for $$x$$ and $$y$$ using these equations and simplify.

 $$x = r \cos(\theta)$$ $$y = r \sin(\theta)$$

Polar → Rectangular Points

In this case, we are given specific $$r$$ and $$\theta$$ values and we want to find the point $$(x,y)$$. This is the easiest direction because we already have the equations in the form we need, i.e. $$x = r \cos(\theta)$$ and $$y = r \sin(\theta)$$. For examples, check out the practice problems.

Polar → Rectangular Equations

Equations - - Here, we are given a function $$g(r,\theta)=0$$ and we need to find an equation $$f(x,y)=0$$. One of the keys to solving these problems is to use the fact $$\cos^2(\theta) + \sin^2(\theta) = 1$$. In terms of the above equations, we have
$$\displaystyle{ \begin{array}{rcl} x^2 + y^2 & = & [ r \cos(\theta) ]^2 + [ r \sin(\theta) ]^2 \\ & = & r^2[ \cos^2(\theta) + r \sin^2(\theta)] \\ & = & r^2 \end{array} }$$

### Converting Resources

use these filters to show only the practice problems you want to see

type ↓ | direction →

rect → polar

polar → rect

points

equations

Practice Problems

Instructions - - Unless otherwise instructed, convert the following points or equations accordingly, giving angles in radians and all other answers in exact terms.
Note: When you are asked to calculate $$g(r,\theta) = 0$$ this means to find the equation in polar form and move all terms to one side of the equal sign for your final answer. Similarly for equations in rectangular form.

 Level A - Basic

Practice A01

given

$$(x,y) = (0,2)$$

calculate

$$( r, \theta )$$

Practice A02

given

$$(x,y) = (-3,4)$$

calculate

$$( r, \theta )$$

Practice A03

given

$$(x,y) = (1,-5)$$

calculate

$$( r, \theta )$$

Practice Problem [ A01 ]

Problem Statement

given

$$(x,y) = (0,2)$$

calculate

$$( r, \theta )$$

Problem Solution

 video by PatrickJMT

$$\displaystyle{ (r, \theta) = (2, \pi/2) }$$

like? 3

Practice Problem [ A02 ]

Problem Statement

given

$$(x,y) = (-3,4)$$

calculate

$$( r, \theta )$$

Problem Solution

 video by PatrickJMT

$$\displaystyle{ (r, \theta) = (5, 2.215 ) }$$

like? 2

Practice Problem [ A03 ]

Problem Statement

given

$$(x,y) = (1,-5)$$

calculate

$$( r, \theta )$$

Problem Solution

 video by PatrickJMT

$$\displaystyle{ (r, \theta) = (\sqrt{26}, -1.373 ) }$$

like? 0

Practice A04

calculate

$$(x,y)$$

given

$$( r, \theta ) = (4,\pi/3)$$

Practice A05

calculate

$$(x,y)$$

given

$$( r, \theta ) = (2,\pi)$$

Practice A06

calculate

$$f(x,y) = 0$$

given

$$r=3\cos(\theta)$$

Practice Problem [ A04 ]

Problem Statement

calculate

$$(x,y)$$

given

$$( r, \theta ) = (4,\pi/3)$$

Problem Solution

 video by PatrickJMT

$$\displaystyle{ (x, y) = (2, 2\sqrt{3}) }$$

like? 0

Practice Problem [ A05 ]

Problem Statement

calculate

$$(x,y)$$

given

$$( r, \theta ) = (2,\pi)$$

Problem Solution

 video by PatrickJMT

$$\displaystyle{ (x, y) = (-2, 0) }$$

like? 2

Practice Problem [ A06 ]

Problem Statement

calculate

$$f(x,y) = 0$$

given

$$r=3\cos(\theta)$$

Problem Solution

 video by PatrickJMT

$$\displaystyle{ x^2-3x+y^2 = 0 }$$

like? 1

Practice A07

given

$$x^2+y^2 = 1$$

calculate

$$g(r,\theta) = 0$$

Practice A08

given

$$y = 2x+1$$

calculate

$$g(r,\theta) = 0$$

Practice A09

given

$$y=3/x$$

calculate

$$g(r,\theta) = 0$$

Practice Problem [ A07 ]

Problem Statement

given

$$x^2+y^2 = 1$$

calculate

$$g(r,\theta) = 0$$

Problem Solution

 video by PatrickJMT

$$\displaystyle{ r - 1 = 0 }$$

like? 2

Practice Problem [ A08 ]

Problem Statement

given

$$y = 2x+1$$

calculate

$$g(r,\theta) = 0$$

Problem Solution

 video by PatrickJMT

$$\displaystyle{ r + \frac{1}{2\cos(\theta)-\sin(\theta)} = 0 }$$

like? 1

Practice Problem [ A09 ]

Problem Statement

given

$$y=3/x$$

calculate

$$g(r,\theta) = 0$$

Problem Solution

 video by PatrickJMT

$$\displaystyle{ r^2 - \frac{3}{\cos(\theta)\sin(\theta)} = 0 }$$

like? 0

Practice A10

calculate

$$(x,y)$$

given

$$(r, \theta) = (1, \pi/4)$$

Practice A11

calculate

$$(x,y)$$

given

$$(r, \theta) = (-2, 2\pi/3)$$

Practice A12

calculate

$$(x,y)$$

given

$$(r, \theta) = (1, -\pi/3)$$

Practice Problem [ A10 ]

Problem Statement

calculate

$$(x,y)$$

given

$$(r, \theta) = (1, \pi/4)$$

Problem Solution

 video by IntegralCALC

$$\displaystyle{ ( \sqrt{2}/2, \sqrt{2}/2 ) }$$

like? 0

Practice Problem [ A11 ]

Problem Statement

calculate

$$(x,y)$$

given

$$(r, \theta) = (-2, 2\pi/3)$$

Problem Solution

 video by IntegralCALC

$$\displaystyle{ (1, -\sqrt{3} ) }$$

like? 0

Practice Problem [ A12 ]

Problem Statement

calculate

$$(x,y)$$

given

$$(r, \theta) = (1, -\pi/3)$$

Problem Solution

 video by IntegralCALC

$$\displaystyle{ ( 1/2, -\sqrt{3}/2 ) }$$

like? 0

Practice A13

calculate

$$(x,y)$$

given

$$(r, \theta) = (3, 3\pi/2)$$

Practice A14

calculate

$$(x,y)$$

given

$$(r, \theta) = (2, -\pi/4)$$

Practice A15

calculate

$$f(x,y)=0$$

given

$$r = -5\cos(\theta)$$

Practice Problem [ A13 ]

Problem Statement

calculate

$$(x,y)$$

given

$$(r, \theta) = (3, 3\pi/2)$$

Problem Solution

 video by IntegralCALC

$$\displaystyle{ ( 0, -3 ) }$$

like? 0

Practice Problem [ A14 ]

Problem Statement

calculate

$$(x,y)$$

given

$$(r, \theta) = (2, -\pi/4)$$

Problem Solution

 video by IntegralCALC

$$\displaystyle{ ( \sqrt{2}, -\sqrt{2} ) }$$

like? 0

Practice Problem [ A15 ]

Problem Statement

calculate

$$f(x,y)=0$$

given

$$r = -5\cos(\theta)$$

Problem Solution

 video by IntegralCALC

$$\displaystyle{ x^2 + 5x + y^2 = 0 }$$

like? 1

Practice A16

given

$$y = x^2$$

calculate

$$g(r, \theta) = 0$$

Practice A17

given

$$xy = 1$$

calculate

$$g(r, \theta) = 0$$

Practice Problem [ A16 ]

Problem Statement

given

$$y = x^2$$

calculate

$$g(r, \theta) = 0$$

Problem Solution

 video by IntegralCALC

$$\displaystyle{ r - \tan(\theta) \sec(\theta) = 0 }$$

like? 0

Practice Problem [ A17 ]

Problem Statement

given

$$xy = 1$$

calculate

$$g(r, \theta) = 0$$

Problem Solution

 video by IntegralCALC

$$\displaystyle{ r^2 - \sec(\theta) \csc(\theta) =0 }$$

like? 0

 Level B - Intermediate

Practice B01

calculate

$$(x,y)$$

given

$$( r, \theta ) = (-7, -2\pi/3)$$

Practice B02

calculate

$$f(x,y) = 0$$

given

$$r^2 = -3\sec(\theta)$$

Practice B03

calculate

$$f(x,y) = 0$$

given

$$r = 3\sin(\theta)\tan(\theta)$$

Practice Problem [ B01 ]

Problem Statement

calculate

$$(x,y)$$

given

$$( r, \theta ) = (-7, -2\pi/3)$$

Problem Solution

 video by PatrickJMT

$$\displaystyle{ (x, y) = (7/2, 7\sqrt{3}/2) }$$

like? 0

Practice Problem [ B02 ]

Problem Statement

calculate

$$f(x,y) = 0$$

given

$$r^2 = -3\sec(\theta)$$

Problem Solution

 video by PatrickJMT

$$\displaystyle{ x^4 + x^2y^2 - 9=0 }$$

like? 0

Practice Problem [ B03 ]

Problem Statement

calculate

$$f(x,y) = 0$$

given

$$r = 3\sin(\theta)\tan(\theta)$$

Problem Solution

 video by PatrickJMT

$$\displaystyle{ x^3+xy^2 -3y^2 = 0}$$

like? 0