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Converting Between Polar and Rectangular Form

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 and rectangular coordinates describe the same point with different anchors. 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

Why it matters: Parametric, polar, vector, and limit ideas prepare students for motion, curves, and the rate-of-change thinking used in Calculus.

Worked example

Problem. Convert the rectangular point (6, 8) to its polar radius r.

  1. Worked Example: First identify exactly what the question is asking: Convert the rectangular point (6, 8) to its polar radius r.
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Use r = sqrt(x^2 + y^2).
  4. r = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100).
  5. So r = 10.
  6. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.

Answer: 10

Practice problems

1. Convert the rectangular point (6, 8) to its polar radius r.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Convert the rectangular point (6, 8) to its polar radius r.
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Use r = sqrt(x^2 + y^2).
  4. r = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100).
  5. So r = 10.
  6. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.

Answer: 10

2. Convert the polar point (r, theta) = (4, 90 degrees) to rectangular form (x, y).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Convert the polar point (r, theta) = (4, 90 degrees) to rectangular form (x, y).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Use x = r cos(theta), y = r sin(theta).
  4. x = 4 cos(90) = 4(0) = 0 and y = 4 sin(90) = 4(1) = 4.
  5. So (x, y) = (0, 4).
  6. Check the result by substituting or estimating: the response should match (0, 4) and make sense in the original problem.

Answer: (0, 4)

3. Which pair of relations converts polar (r, theta) to rectangular (x, y)?

Choices: x = r cos(theta), y = r sin(theta) · x = r sin(theta), y = r cos(theta) · x = r + theta, y = r - theta · x = r/cos(theta), y = r/sin(theta)

Show solution
  1. The horizontal leg of the reference triangle is the adjacent side, r cos(theta).
  2. The vertical leg is the opposite side, r sin(theta).
  3. So x = r cos(theta) and y = r sin(theta).

Answer: x = r cos(theta), y = r sin(theta)

4. Convert the polar point (2, 0 degrees) to rectangular form (x, y).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Convert the polar point (2, 0 degrees) to rectangular form (x, y).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use x = r cos(theta), y = r sin(theta).
  4. x = 2 cos(0) = 2(1) = 2 and y = 2 sin(0) = 2(0) = 0.
  5. So (x, y) = (2, 0).
  6. Check the result by substituting or estimating: the response should match (2, 0) and make sense in the original problem.

Answer: (2, 0)

5. A point is given in polar form as (3, 180 degrees). Rewrite it as a rectangular point (x, y).

Show solution
  1. Core Practice: First identify exactly what the question is asking: A point is given in polar form as (3, 180 degrees). Rewrite it as a rectangular point (x, y).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use x = r cos(theta), y = r sin(theta).
  4. x = 3 cos(180) = 3(-1) = -3 and y = 3 sin(180) = 3(0) = 0.
  5. So (x, y) = (-3, 0).
  6. Check the result by substituting or estimating: the response should match (-3, 0) and make sense in the original problem.

Answer: (-3, 0)

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