Polar Coordinates and Their Graphs

Definition

Polar coordinates offer a way to represent points in a plane by using a distance from a fixed point and an angle from a fixed direction. Instead of the familiar (x, y) Cartesian coordinates, polar coordinates are written as (r, θ), where ‘r’ is the distance from the origin (pole) and ‘θ’ is the angle measured from the positive x-axis.

Who

Some of the common fields that have the application of polar coordinates include the following.

  • Sonar and Radar Systems
    • Polar coordinates are not only used in sonar and radar systems but also in functions such as navigation and mapping objects underwater and in the air. Surveillance and radar units utilize polar coordinates, along with distances and angles from the emitting source, to pinpoint the target’s location.
    • Radar systems, used in air traffic control and other fields, utilize polar coordinates to determine the position and movement of objects based on their distance and angle from the radar station.
  • Navigation and GPS Systems
    • Polar coordinates are widely used in navigation and GPS (Global Positioning System) for determining locations on Earth. GPS receivers use the polar coordinates (latitude and longitude) to pinpoint the exact geographic position of airplanes, ships, and vehicles. This enables accurate navigation and route planning.
    • Aircraft and ships use polar coordinates or slight variations of them to specify directions and distances from a reference point.
  • Engineering and Robotics
    • In engineering and robotics, describing an object’s position in space using polar coordinates is very important because it helps analyze and implement an object’s orientation. Engineers use polar coordinates to control robot arms and carry out automation processes to ensure precise movements and successful manipulation tasks in industries.
    • When working with rotating objects or systems (such as gears, pendulums, or rotating machinery), polar coordinates make calculations of speed, torque, and other rotational parameters easier.
  • Physics and Astronomy
    • Polar coordinates are very important in physics and astronomy for identifying the motion of objects like planets, stars, and satellites in space. The trajectories of particles in accelerators and the movement of celestial bodies are constructed in polar coordinates, which are used by physicists.
    • Polar coordinates are especially helpful for describing fields that spread out from a central point, such as magnetic or gravitational fields.
    • Astronomers use polar coordinates (often in a spherical form) to predict the positions of celestial objects.
  • Biology and Biomedical Imaging
    • Polar coordinates are useful tools in both biology and biomedical imaging, helping analyze biological structures and medical images. Scientists use polar coordinates to study cell morphology, tissue structures, and biological patterns in microscopy and medical techniques.
    • Some imaging methods, such as sonar, use polar coordinates to create and visualize data in a circular or radial pattern.
  • Electrical Engineering and Signal Processing
    • The polar coordinates theory is used in electrical engineering and signal processing to represent complex numbers and analyze AC circuits. Electrical engineers use polar coordinates to depict phasors, which help analyze the amplitude and phase of AC voltages and currents.
    • Polar coordinates are useful for representing signals, especially sound or radio waves, which often have radial characteristics.

What

Polar coordinates are a two-dimensional coordinate system used to locate points in a plane. Unlike the Cartesian coordinate system, which uses perpendicular axes (x and y) to represent points, the polar coordinate system uses a radial distance and an angle.

 

In polar coordinates, a point is represented by an ordered pair (r, θ), where:

  • Pole: The fixed reference point in the polar coordinate system.
  • Polar Axis: The ray (half-line) from the pole, typically aligned with the positive x-axis.
  • r (Radial Coordinate): The distance from the pole to the point (radius).
  • θ (Angular Coordinate): The angle measured counterclockwise from the polar axis (normally the positive x-axis) to the line segment which connects origin and the point (polar angle).

Why

Data should be graphed in polar coordinates when the phenomenon being studied or the data itself is inherently tied to a central point, direction, and distance.

How to Read a Polar Graph? – PlotDigitizer

 

See Theoretical Knowledge Vs Practical Application.

How

Conversion Formulas

  • Cartesian to Polar
    • r = √(x² + y²)
    • θ = tan⁻¹(y/x) (Be mindful of the quadrant to get the correct angle)
  • Polar to Cartesian
    • x = r * cos(θ)
    • y = r * sin(θ)
Aspect
Cartesian Coordinates
Polar Coordinates
Dimension
Can be used in 2D and 3D (x, y, z)
Primarily used in 2D (r, θ)
Representation
Points are located by their distance from two perpendicular axes (x and y)
Points are located by their distance from a fixed origin (r) and angle from a fixed reference direction (θ)
Visualization
Grid of horizontal and vertical lines
Origin with concentric circles radiating outward and lines at regular angular intervals
Strengths
Simple and intuitive for rectangular shapes and straight lines
Efficient for representing circular and radial relationships
Weaknesses
Can be cumbersome for circular shapes and angles
Can have ambiguity for certain points (e.g., negative radii)
Applications
Plotting points, calculating distances and areas, linear equations, motion in straight lines
Describing circular motion, planetary orbits, wave patterns, polar plots
Example
Plotting the location of a house on a map (x, y)
Describing the position of a planet around the sun (r, θ)

Polar vs Cartesian Coordinates – GeeksforGeeks

When to Graph in Polar Coordinates

Polar coordinates are most useful for graphing data that has a natural circular or radial relationship to a central point. They are especially helpful when working with spirals, circular patterns, or phenomena involving angles and distances from a central location. Plotting data in polar coordinates can provide a more natural and insightful representation than Cartesian coordinates.

  • Circular or Radial Symmetry: When data points are arranged around a central point, or when the relationships are best described by distances and angles from a center, polar coordinates can simplify the representation and analysis.
  • Spirals and Rotational Data: Data that forms spirals, like the path of a spinning object or the movement of particles in a swirling field, are naturally suited to polar coordinates.
  • Situations Involving Angles and Distances: If the data involves angles (like directions or bearings) and distances from a fixed point, polar coordinates provide a more intuitive way to represent and understand relationships.
Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind. How can the sailor indicate his location to the Coast Guard? – lumencandela

Graphing Polar Coordinates

To graph polar coordinates, understand that they are written as (r, θ), where ‘r’ is the radius (distance from the origin and ‘θ’ is the angle from the positive x-axis. First, visualize or draw the angle θ on the polar plane. Then, move along that angle’s ray a distance of ‘r’ units from the origin (pole). If ‘r’ is negative, move in the opposite direction along the extended ray.

  1. Locate the angle θ: Start at the polar axis and rotate counterclockwise (if θ is positive) or clockwise (if θ is negative) by the specified angle.
  2. Locate the radius r: From the pole, move along the ray representing the angle θ. If r is positive, move outward from the pole. If r is negative, move inward along the line extended from the pole in the opposite direction of the angle.
  3. Plot the point: Mark the point where the radius intersects the angle’s ray.
Polar coordinates: Definitions – xaktly

 

Many of the References and Additional Reading websites and Videos will assist you with understanding and applying polar coordinates and their graphs.

As some professors say: “It is intuitively obvious to even the most casual observer.“

References

“‌How to Read a Polar Graph?” 2021. January 17. https://plotdigitizer.com/how-read-polar-plotdigitizer.

Steketee, Scott. 2013. “Cartesian and Polar Graphs | Sine of the Times.” Sine of the Times | the Web Sketchpad and Math Education Blog. May 2013. https://www.sineofthetimes.org/cartesian-and-polar-graphs/.

“Polar Coordinates | Theory.” 2025. datacamp.com. 2025. https://campus.datacamp.com/courses/understanding-data-visualization/99-problems-but-a-plot-aint-one-of-them?ex=1.

‌“Polar Coordinate System.” 2019. BYJUS. BYJU’S. December 11, 2019. https://byjus.com/maths/polar-coordinates/.

‌“Spherical Coordinates | Calculus III.” 2025. lumenlearning.com. 2025. https://courses.lumenlearning.com/calculus3/chapter/spherical-coordinates/.

‌“Polar Coordinates | Algebra and Trigonometry.” 2025. lumenlearning.com. 2025. https://courses.lumenlearning.com/suny-osalgebratrig/chapter/polar-coordinates/.

‌“Polar Coordinate System.” 2025. Wikipedia. Wikimedia Foundation. July 29. https://en.wikipedia.org/wiki/Polar_coordinate_system.

“GPS Coordinate Conversion.” 2025. msoe.edu. 2025. https://faculty-web.msoe.edu/hornick/Courses/se2800/labs/GPS%20Cooordinate%20Conversion.html.

‌CK-12 Foundation. 2025. “CK12-Foundation.” CK-12 Foundation. 2025. https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/10.2/primary/lesson/polar-equations-of-conics-pcalc/.

‌“Polar Charts 101: An In-Depth Guide.” 2025. inforiver. June 4, 2025. https://inforiver.com/insights/polar-charts-101/.

‌“Area and Arc Length in Polar Coordinates | Calculus II Class Notes | Fiveable.” 2024. fiveable. 2024. https://library.fiveable.me/calc-ii/unit-7/4-area-arc-length-polar-coordinates/study-guide/F1A2DVeDPIh2mnT4.

‌“Polar Coordinate System and Transformation | Calculus IV Class Notes | Fiveable.” 2024. fiveable. 2024. https://library.fiveable.me/calculus-iv/unit-11/polar-coordinate-system-transformation/study-guide/hNpGQa0LcAsNvTZG.

‌“11.3: Polar Coordinates.” 2016. Mathematics LibreTexts. July 11, 2016. https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/11%3A_Parametric_Equations_and_Polar_Coordinates/11.03%3A_Polar_Coordinates.

‌“8.1: Polar Coordinates and Polar Graphs.” 2018. Mathematics LibreTexts. December 22, 2018. https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/08%3A_Some_Curvilinear_Coordinate_Systems/8.01%3A_Polar_Coordinates_and_Polar_Graphs.

‌“6.2: Graphing Basic Polar Equations.” 2021. Mathematics LibreTexts. April 21, 2021. https://math.libretexts.org/Courses/Rio_Hondo/Math_175%3A_Plane_Trigonometry/06%3A_The_Polar_System/6.02%3A_Graphing_Basic_Polar_Equations.

‌“1.8: Cylindrical and Spherical Coordinates.” 2020. Mathematics LibreTexts. January 17, 2020. https://math.libretexts.org/Courses/University_of_Maryland/MATH_241/01%3A_Vectors_in_Space/1.08%3A_Cylindrical_and_Spherical_Coordinates.

‌“Polar Coordinates – Math Insight.” 2025. mathinsight.org. 2025. https://mathinsight.org/polar_coordinates.

‌Strang, Gilbert, and Edwin Jed Herman. 2016. “7.3 Polar Coordinates – Calculus Volume 2 | OpenStax.” openstax.org. OpenStax. 2016. https://openstax.org/books/calculus-volume-2/pages/7-3-polar-coordinates.

‌Baumback, Wally. 2021. “Module 10: Drawing Lines Using Polar Coordinates.” opentextbc.ca. BCcampus. November 3, 2021. https://opentextbc.ca/autocad2d/chapter/drawing-lines-using-polar-coordinates/.

‌“3.2: Coordinate Systems.” 2020. Physics LibreTexts. April 27, 2020. https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Dourmashkin)/03%3A_Vectors/3.02%3A_Coordinate_Systems.

‌“How to Plot Points in Polar Coordinates.” 2025. study.com. 2025. https://study.com/skill/learn/how-to-plot-points-in-polar-coordinates-explanation.html.

‌McMullin, Lin. 2024. “Why Polar Equations?” Teaching Calculus. January 23, 2024. https://teachingcalculus.com/2024/01/23/why-polar-equations/.

‌“Calculus II – Polar Coordinates.” 2025. lamar.edu. 2025. https://tutorial.math.lamar.edu/classes/calcii/polarcoordinates.aspx.

‌Team, The Albert. 2025. “Graphing Polar Functions: A Complete AP® Precalculus Review | Albert Resources.” Albert Resources. March 7, 2025. https://www.albert.io/blog/graphing-polar-functions-a-complete-ap-precalculus-review/.

‌“Comparing Coordinate Systems: Reflect on Your Experiences with Cartesian and Polar Coordinates. How Would You Explain…” 2025. cliffsnotes.com. March 11, 2025. https://www.cliffsnotes.com/cliffs-questions/3117743.

‌“How to Plot Polar Coordinates | Dummies.” 2021. dummies.com. 2021. https://www.dummies.com/article/academics-the-arts/math/pre-calculus/how-to-plot-polar-coordinates-167586/.

‌“Polar Coordinate Systems | EBSCO.” 2022. EBSCO Information Services, Inc. | http://www.ebsco.com. 2022. https://www.ebsco.com/research-starters/history/polar-coordinate-systems.

‌GeeksforGeeks. 2024. “RealLife Applications of Polar Coordinates.” GeeksforGeeks. April 2024. https://www.geeksforgeeks.org/maths/real-life-applications-of-polar-coordinates/.

‌“Polar and Cartesian Coordinates.” 2025. Math Is Fun. 2025. https://www.mathsisfun.com/polar-cartesian-coordinates.html.

‌Lee, Sarah. 2025. “Practical Applications of Polar Coordinates.” numberanalytics.com. 2025. https://www.numberanalytics.com/blog/polar-coordinates-applications-calculus.

‌Lee, Sarah. 2025. “Polar Coordinates Simplified.” numberanalytics.com. 2025. https://www.numberanalytics.com/blog/polar-coordinates-simplified-pre-calculus.

‌Lee, Sarah. 2025. “The Ultimate Polar Graphing Guide.” numberanalytics.com. 2025. https://www.numberanalytics.com/blog/ultimate-polar-graphing-guide.

‌Bogna Szyk. 2017. “Polar Coordinates Calculator.” Omni Calculator. March 10, 2017. https://www.omnicalculator.com/math/cartesian-to-polar.

‌“Intro to Polar Coordinates.” 2024. pearson.com. 2024. https://www.pearson.com/channels/precalculus/learn/patrick/15-polar-equations/polar-coordinate-system.

‌Academy, Queen Elizabeth. 2025. “Understanding and Using Polar Coordinates | Queen Elizabeth Academy.” Queen Elizabeth Academy. January 4. https://www.qetutoring.com/polar-coordinates.html.

‌Summers, Vincent. 2014. “Polar Coordinates – an Introduction with Examples.” Quirky Science. November 17. https://www.quirkyscience.com/introduction-polar-coordinates/.‌

Additional Reading

“11.5: Graphs of Polar Equations.” 2021. Mathematics LibreTexts. August 2021. https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/11%3A_Applications_of_Trigonometry/11.05%3A_Graphs_of_Polar_Equations.

Bourne, Murray. 2009. “Polar Coordinates and Cardioid Microphones – Interactive Mathematics.” Interactive Mathematics – Mathematics, Learning, Computing, Travel – and Whatever… October 18. https://www.intmath.com/blog/mathematics/polar-coordinates-and-cardioid-microphones-2496.

Cook, John. 2023. “Lemniscate Functions.” John D. Cook | Applied Mathematics Consulting. October 4. https://www.johndcook.com/blog/2023/10/04/lemniscate-functions/.

‌‌‌‌GeeksforGeeks. 2021. “Area between Polar Curves.” GeeksforGeeks. July 11, 2021. https://www.geeksforgeeks.org/maths/area-between-polar-curves/.

GeeksforGeeks. 2024. “Polar Coordinates System.” GeeksforGeeks. January 11, 2024. https://www.geeksforgeeks.org/maths/polar-coordinates-system/.

‌GeeksforGeeks. 2021. “Polar Representation of Complex Numbers.” GeeksforGeeks. March 2, 2021. https://www.geeksforgeeks.org/maths/polar-representation-of-complex-numbers/.

GeeksforGeeks. 2024. “Real Life Applications of Cardioid.” GeeksforGeeks. May 7, 2024. https://www.geeksforgeeks.org/maths/real-life-applications-of-cardioid/.

Kirvan, Paul. 2022. “GPS Coordinates.” WhatIs. TechTarget. 2022. https://www.techtarget.com/whatis/definition/GPS-coordinates.

‌‌Lee, Frank and John Eckhart. “Counting Rose Petals.” 2021. Plus Maths. April 29, 2021. https://plus.maths.org/content/counting-rose-petals.

Lee, Sarah. 2025. “Lemniscate Graphing in Polar Coordinates.” numberanalytics.com. 2025. https://www.numberanalytics.com/blog/lemniscate-polar-graphing-algebra-ii.

‌“Maths in Excel – 2D Polar Plots.” 2015. simplexct.com. 2015. https://simplexct.com/maths-in-excel-graph-2d-polar-plots-in-excel.

‌‌“Polar Coordinate Graph.” 2024. Statspaces. 2024. https://statspaces.weebly.com/blog/polar-coordinate-graph.

“Polar Power.” 2014. Plus Maths. April 3, 2014. https://plus.maths.org/content/polar-power.

Stempel, Philip. “What Is a Cardioid Polar Pattern?” 2025. LEWITT. 2025. https://www.lewitt-audio.com/blog/cardioid-polar-pattern.

Vedantu. 2020. “Cardioid – Definition, Equation, Properties & Examples.” VEDANTU. Vedantu. June 26, 2020. https://www.vedantu.com/maths/cardioid.

‌“How Do I Graph Cardioid r = a(1 – cosθ)?” 2021. VEDANTU. Vedantu.com. January 8, 2021. https://www.vedantu.com/question-answer/graph-cardioid-raleft-1cos-theta-right-class-11-maths-cbse-5ff84cf61eb9072eb80c733d.

‌‌‌Willard, David. 2018. “Understanding Microphone Polar Patterns – Azden.” Azden. February 9, 2018. https://www.azden.com/blog/understanding-microphone-polar-patterns/.‌

Videos

 

 

Medium Member Only Medium Member Only

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.

The featured image on this page is from the Sine of the Times website.

The model shows the function f(θ) = 1 – cos 2θ in both Cartesian and polar form. For each graph, the independent variable appears as a red bar that corresponds to a particular value of x (for Cartesian) or θ (for polar). The red bar has tick marks that show possible values of the dependent variable (y for Cartesian and r for polar), and a green bow-tie that shows the actual value of the dependent variable for this value of the independent variable.

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