Polar Coordinates
Locate a point by its distance r and angle θ.
About this topic
Polar coordinates locate a point by its distance r from the origin and its angle θ from the positive x-axis, written (r, θ). Conversions: x = r cos θ, y = r sin θ, and r² = x² + y². A point has many polar representations.
Some curves are simple in polar form: r = 3 is a circle, θ = π/4 is a line through the origin, r = 1 + cos θ is a heart-shaped cardioid and r = 2 sin 3θ is a three-petalled rose. Negative r means going in the opposite direction, and adding 2π to the angle gives the same point.
To convert a point such as (−1, 1) to polar form, r = √2 and tan θ = −1, but the point is in quadrant II, so θ = 3π/4 rather than the −π/4 a calculator gives. Always check the quadrant.
Polar coordinates suit radar, sonar, navigation, antenna patterns, spirals in nature and anything with rotational symmetry. They also simplify area and integration problems in calculus. A common error is using arctan blindly without adjusting for the quadrant.
Key ideas
- x = r cos θ, y = r sin θ
- r = √(x² + y²)
- Polar curves include circles and spirals
Quick check
Use cases: where Polar Coordinates helps
- Homework: review the key idea behind Polar Coordinates before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Polar Coordinates without notes.
- Visual learning: play with the interactive lab to see how Polar Coordinates changes when you adjust its inputs.
- Teaching a friend: explain Polar Coordinates in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Polar Coordinates?
Locate a point by its distance r and angle θ.
What will I learn in Polar Coordinates?
x = r cos θ, y = r sin θ. r = √(x² + y²). Polar curves include circles and spirals.
Is Polar Coordinates part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Polar Coordinates lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Polar Coordinates?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.