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4.1: Polar Coordinates

Difficulty Level: At Grade Created by: CK-12
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Everyone has dreamed of flying at one time or another. Not only would there be much less traffic to worry about, but directions would be so much simpler!

Walking or driving: "Go East 2 blocks, turn left, then North 6 blocks. Wait for the train. Turn right, East 3 more blocks, careful of the cow! Turn left, go North 4 more blocks and park."

Flying: "Fly 30deg East of North for a little less than 11 and 1/4 blocks. Land."

Nice daydream, what does it have to do with polar coordinates?

Watch This

Embedded Video:

- Khan Academy: Polar Coordinates 1


The Polar Coordinate System is alternative to the Cartesian Coordinate system you have used in the past to graph functions. The polar coordinate system is specialized for visualizing and manipulating angles.

Angles are identified by travelling counter-clockwise around the circular graph from the 0deg line, or r-axis (where the + x axis would be) to a specified angle.

To plot a specific point, first go along the r-axis by r units. Then, rotate counterclockwise by the given angle, commonly represented "θ". Be careful to use the correct units for the angle measure (either radians or degrees).


Usually polar plots are done with radians (especially if they include trigonometric functions), but sometimes degrees are used.

A radian is the angle formed between the r axis and a polar axis drawn to meet a section of the circumference that is the same length as the radius of a circle.

Given that the circumference of a circle is \begin{align*}2 \pi \cdot r\end{align*}2πr, and since r is the radius, that means there are \begin{align*}2 \pi\end{align*}2π radians in a complete circle, and \begin{align*}1 \pi\end{align*}1π radians in 1/2 of a circle.

If 1/2 of a circle is \begin{align*}\pi\end{align*}π radians, and is 180deg, that means that there are \begin{align*}\frac{180}{\pi}\end{align*}180π degrees in each radian.

That translates to approximately 57.3 degrees = 1 radian.

Graphing using technology

Polar Equations can be graphed using a graphing calculator: With the graphing calculator- go to MODE. There select RADIAN for the angle measure and POL (for Polar) on the FUNC (function)line. When Y = is pressed, note that the equation has changed from y = to r = . There input the polar equation. After pressing graph, if you can’t see the full graph, adjust x- and y- max/min, etc in WINDOW.

Example A

Plot the points on a polar coordinate graph:

Point A \begin{align*}\left (2,\frac{\pi}{3}\right )\end{align*}(2,π3)

Point B \begin{align*}(4, 135^{o})\end{align*}(4,135o)

Point C \begin{align*}\left (-2,\frac{\pi}{6}\right )\end{align*}(2,π6)


Below is the pole, polar axis and the points A, B and C.

Example B

Plot the following points:

a. \begin{align*}(4, 30^o)\end{align*}(4,30o)

b. \begin{align*}(2.5, \pi)\end{align*}(2.5,π)

c. \begin{align*}\left(-1,\frac{\pi}{3}\right )\end{align*}(1,π3)

d. \begin{align*}\left(3,\frac{5\pi}{6}\right )\end{align*}(3,5π6)

e. \begin{align*}(-2, 300^o)\end{align*}(2,300o)


Example C

Use a graphing calculator or plotting program to plot the following equations:

a. \begin{align*}r = 1 + 3 sin \theta\end{align*}r=1+3sinθ

b. \begin{align*}r = 1 + 2 cos \theta\end{align*}r=1+2cosθ




Review the steps above under Graphing using technology if you are having trouble.


The polar coordinate system is a specialized graph used for angles and angle manipulations.

The pole is the center point on a polar graph.

One radian is the angle formed by moving counter-clockwise around the circumference of a circle by the length of the radius. It is equal to apx 57.3 degrees.

The polar axis is a ray drawn from the pole at the 0o angle on a polar graph.

Guided Practice

1) Plot the points on a polar graph:

a) \begin{align*}\left(2, \frac{\pi}{3}\right)\end{align*}(2,π3)
b) \begin{align*}(3, 90^o)\end{align*}(3,90o)
c) \begin{align*}(1.5, \pi)\end{align*}(1.5,π)

2) Convert from radians to degrees:

a) \begin{align*}\frac{\pi}{2}\end{align*}π2
b) \begin{align*}5.17\end{align*}5.17
c) \begin{align*}\frac{3\pi}{2}\end{align*}3π2

3) Convert from degrees to radians:

a) \begin{align*}251^o\end{align*}251o
b) \begin{align*}360^o\end{align*}360o
c) \begin{align*}327^o\end{align*}327o

4) Convert from degrees to radians, answer in terms of \begin{align*}\pi\end{align*}π:

a) \begin{align*}90^o\end{align*}90o
b) \begin{align*}270^o\end{align*}270o
c) \begin{align*}45^o\end{align*}45o


1) The points are plotted on the graph below:

2) Recall that \begin{align*}\pi rad = 180^o\end{align*}πrad=180o and \begin{align*}1rad = \frac{180}{\pi} \approx 57.3^o\end{align*}1rad=180π57.3o

a) If \begin{align*}\pi rad = 180^o\end{align*}πrad=180o then \begin{align*}\frac{\pi}{2}rad = 90^o\end{align*}π2rad=90o
b) If \begin{align*}1rad \approx 57.3^o\end{align*}1rad57.3o then \begin{align*}5.17rad \approx 296^o\end{align*}5.17rad296o
c) If \begin{align*}\pi rad = 180^o\end{align*}πrad=180o then \begin{align*}\frac{3\pi}{2}rad = 270^o\end{align*}3π2rad=270o

3) Recall that \begin{align*}\frac{180^o}{\pi} = 57.3^o \approx 1rad\end{align*}180oπ=57.3o1rad

a) If \begin{align*}57.3^o \approx 1rad\end{align*}57.3o1rad then \begin{align*}251^o \approx 4.38rad \approx 1.4\pi rad\end{align*}251o4.38rad1.4πrad
b) If \begin{align*}57.3^o \approx 1rad\end{align*}57.3o1rad then \begin{align*}360^o \approx 6.28rad\end{align*}360o6.28rad
c) If \begin{align*}57.3^o \approx 1rad\end{align*}57.3o1rad then \begin{align*}\frac{327^o}{57.3^o} \approx 5.71 rad\end{align*}327o57.3o5.71rad

4) Recall that \begin{align*}2\pi rad = 360^o\end{align*}2πrad=360o and therefore \begin{align*}\pi rad = 180^o\end{align*}πrad=180o

a) If \begin{align*}\pi rad = 180^o\end{align*}πrad=180o then \begin{align*}\frac{\pi}{2} rad = 90^o\end{align*}π2rad=90o
b) If \begin{align*} \pi rad = 180^o\end{align*}πrad=180o and \begin{align*}\frac{\pi}{2} rad = 90^o\end{align*}π2rad=90o then \begin{align*}1\frac{1}{2}\pi rad \to \frac{3}{2}\pi \to \frac{3\pi}{2} rad = 270^o\end{align*}
c) If \begin{align*}\frac{\pi}{2}rad = 90^o\end{align*} then \begin{align*}\frac{\pi}{4} rad = 45^o\end{align*}


  1. Why can a point on the plane not be labeled using a unique ordered pair \begin{align*}(r, \theta)\end{align*}
  2. Explain how to graph \begin{align*}(r, \theta)\end{align*} if \begin{align*}r < 0\end{align*} and/or \begin{align*}\theta > 360\end{align*}

Graph Each Point in the Polar Plane

  1. A \begin{align*}(6, 145^o)\end{align*}
  2. B \begin{align*}\left(-2, \frac{13\pi}{6} \right)\end{align*}
  3. C \begin{align*}\left(\frac{7}{4}, -210^o\right)\end{align*}
  4. D \begin{align*}\left(5, \frac{\pi}{2}\right)\end{align*}
  5. E \begin{align*}\left(3.5, \frac{-\pi}{8}\right)\end{align*}

Name Two Other Pairs of Polar Coordinates for Each Point

  1. \begin{align*}(1.5, 170^o)\end{align*}
  2. \begin{align*}\left(-5, \frac{\pi}{-3}\right)\end{align*}
  3. \begin{align*}(3, 305^o)\end{align*}

Graph Each Polar Equation

  1. \begin{align*}r = 3\end{align*}
  2. \begin{align*}\theta = \frac{\pi}{5}\end{align*}
  3. \begin{align*}r = 15.5\end{align*}
  4. \begin{align*}r = 1.5\end{align*}
  5. \begin{align*}\theta = -175^o\end{align*}

Find the Distance Between Points

  1. \begin{align*}P_1 \left(5, \frac{\pi}{2}\right)\end{align*} and \begin{align*}P_2 \left(7, \frac{3\pi}{9}\right)\end{align*}
  2. \begin{align*}P_1 (1.3, -52^o) \end{align*} and \begin{align*}P_2 (-13.6, -162^o)\end{align*}
  3. \begin{align*}P_1 (3, 250^o) P_2 (7, 90^o)\end{align*}

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\pi (Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.

Cartesian coordinate system

The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.

polar axis

The polar axis is a ray drawn from the pole at the 0^\circ angle on a polar graph.

polar coordinate system

The polar coordinate system is a special coordinate system in which the location of each point is determined by its distance from the pole and its angle with respect to the polar axis.


The pole is the center point on a polar graph.


A radian is a unit of angle that is equal to the angle created at the center of a circle whose arc is equal in length to the radius.

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Difficulty Level:
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Date Created:
Nov 01, 2012
Last Modified:
May 26, 2016
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