Skill: polar-coordinates

Polar Coordinates & Graphs

Navigate by angle and radius: plot (r, θ) including negative r, convert both ways, and recognize circles, cardioids, and roses.

1 Plotting (r, θ)

Polar coordinates name a point by how far (r) and which direction (θ, measured from the positive x-axis).

Plotting rule

Face the angle θ, then walk r steps — backwards if r is negative

A negative r goes opposite the angle’s ray: (−3, 90°) lands at (0, −3), not (0, 3).
Degrees and radians both work; this course uses degrees for plotting and radians where noted.

Plot (4, 135°) and (−2, 30°)

(4, 135°): face 135° (second quadrant), walk 4 → (−2√2, 2√2) ≈ (−2.83, 2.83).

(−2, 30°): face 30° but walk backwards 2 → (−√3, −1) ≈ (−1.73, −1).

2 Converting between polar and rectangular

The conversion is a right triangle: r is the hypotenuse, θ the angle, x and y the legs.

Conversion triangles

Polar → rectangular: x = r cos θ, y = r sin θ
Rectangular → polar: r = √(x² + y²), tan θ = y/x (use atan2 for the right quadrant)

Keep the triangle handy: x = r cos θ, y = r sin θ, r² = x² + y².
For (x, y) → (r, θ), atan2(y, x) lands θ in the correct quadrant automatically.

Convert (6, 120°) to rectangular

x = 6 cos 120° = 6(−1/2) = −3; y = 6 sin 120° = 6(√3/2) = 3√3 ≈ 5.20. So (−3, 5.20).

Convert (−4, 4) to polar

r = √(16 + 16) = 4√2 ≈ 5.66; θ = atan2(4, −4) = 135°. So (5.66, 135°).

3 Recognizing polar graphs

A few families cover most polar curves. Learn their signatures.

The gallery

r = a → circle centered at the origin, radius |a|
r = a cos θ → circle through the origin (diameter |a| on the x-axis)
r = a ± b cos θ → limacon; a = b gives a cardioid (heart)
r = a cos(nθ) → rose: n petals if n is odd, 2n petals if n is even

Roses are the favorite test question: count the petals from n, not from a sketch.

4 Worked examples

Plot, convert, and classify — each claim verified independently.

Example 1 Polar to rectangular

Convert (5, 90°) to rectangular.

x = 5 cos 90° = 0, y = 5 sin 90° = 5: (0, 5).

Example 2 Polar to rectangular

Convert (4, 180°) to rectangular.

x = 4 cos 180° = −4, y = 4 sin 180° = 0: (−4, 0).

Example 3 Negative r

Convert (−3, 90°) to rectangular.

Negative r goes opposite the 90° ray: x = −3 cos 90° = 0, y = −3 sin 90° = −3: (0, −3).

Example 4 Rectangular to polar

Convert (0, 5) to polar (degrees).

r = 5, θ = 90°: (5, 90°).

Example 5 Rectangular to polar

Convert (−3, 0) to polar (degrees).

r = 3, θ = 180°: (3, 180°).

Example 6 Rose petals

How many petals does r = 4 cos(3θ) have?

n = 3 is odd → 3 petals.

Example 7 Rose petals

How many petals does r = 2 sin(4θ) have?

n = 4 is even → 2·4 = 8 petals.

5 Common mistakes

Two errors that break polar problems.

Plotting negative r in the same direction

Wrong: plotting (−3, 90°) at (0, 3).
Right: negative r goes opposite the angle’s ray: (−3, 90°) lands at (0, −3).

Forgetting r² = x² + y² when converting

Wrong: converting (3, 4) with r = 3 + 4 = 7.
Right: r is a hypotenuse: r = √(3² + 4²) = 5. Keep the triangle x = r cos θ, y = r sin θ, r² = x² + y² handy.

6 Key vocabulary

Words to know

Pole — the origin in polar coordinates.
Polar axis — the positive x-axis, where θ = 0°.
Cardioid — the heart-shaped limacon r = a ± a cos θ.
Rose — r = a cos(nθ): n petals if n odd, 2n if n even.
Limacon — r = a ± b cos θ (dimpled or looped depending on a/b).

7 Quick check

Try these before moving on — click to reveal each answer.

Convert (8, 270°) to rectangular.

x = 8 cos 270° = 0, y = 8 sin 270° = −8: (0, −8).

Convert (−6, 0) to polar (degrees).

r = 6, θ = 180°: (6, 180°).

How many petals does r = 5 sin(2θ) have? What kind of graph is r = 4 + 4 sin θ?

4 petals (n = 2 even); r = 4 + 4 sin θ is a cardioid.

8 Next steps

Now drill the skill with endless randomized problems.

Practice
Try the Tool
Explore the Lab
Print a Worksheet
Play the Game