Polar Coordinates
A free Precalculus lesson from the “Parametric, Polar, Vectors, and Intro to Limits” unit, with a worked example and practice problems including step-by-step solutions.
Polar coordinates describe a point by distance from the origin and direction from the positive x-axis. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.
What you'll learn
- Locate points using radius and angle
- Use polar coordinates in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. In the polar point (3, 90 degrees), what does the first number 3 represent?
- A polar point is written (r, theta).
- The first coordinate r = 3 is the distance from the origin (the pole).
- The second coordinate is the direction angle.
Answer: the distance from the origin
Practice problems
1. In the polar point (3, 90 degrees), what does the first number 3 represent?
Choices: the distance from the origin · the angle of rotation · the slope of the line · the x-coordinate
Show solution
- A polar point is written (r, theta).
- The first coordinate r = 3 is the distance from the origin (the pole).
- The second coordinate is the direction angle.
Answer: the distance from the origin
2. In the polar point (4, 180 degrees), what does the second number 180 degrees represent?
Choices: the direction measured from the positive x-axis · the distance from the origin · the y-coordinate · the radius squared
Show solution
- A polar point is written (r, theta).
- The second coordinate theta = 180 degrees is the direction angle from the positive x-axis.
- The first coordinate r is the distance.
Answer: the direction measured from the positive x-axis
3. In a polar point (r, theta), which coordinate gives the distance from the origin?
Choices: r · theta · the sum r + theta · the product r*theta
Show solution
- Polar coordinates record distance and direction.
- The radius r is the distance from the origin.
- The angle theta is only the direction.
Answer: r
4. A point lies at distance 2 from the origin in the direction 0 degrees. What is its distance from the origin?
Show solution
- Core Practice: First identify exactly what the question is asking: A point lies at distance 2 from the origin in the direction 0 degrees. What is its distance from the origin?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- The distance from the origin is the polar radius r.
- Here r = 2.
- So the distance from the origin is 2.
- Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.
Answer: 2
5. Given the polar point (5, 90 degrees), state its distance from the origin.
Show solution
- Core Practice: First identify exactly what the question is asking: Given the polar point (5, 90 degrees), state its distance from the origin.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- In (r, theta) the first coordinate r is the distance from the origin.
- Here r = 5.
- So the distance from the origin is 5.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
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