Polar Coordinates
Grade 12 · Math · Study Guide
Understand the polar coordinate system and how to convert between polar and rectangular forms.
In the polar coordinate system, a point is described by (r, θ) where r is the distance from the origin and θ is the angle from the positive x-axis. Unlike Cartesian coordinates, many curves have much simpler polar equations.
Conversion between systems uses: x = r cos θ, y = r sin θ, r² = x² + y², and tan θ = y/x. These formulas let you move freely between representations.
Common polar curves include circles (r = a), cardioids (r = a(1 + cos θ)), limaçons, rose curves (r = a cos(nθ)), and lemniscates. The shape depends on the equation structure.
| From Polar | Formula |
|---|---|
| x | r cos θ |
| y | r sin θ |
| r² | x² + y² |
| tan θ | y / x |
| Curve | Equation |
|---|---|
| Circle | r = a |
| Cardioid | r = a(1 + cos θ) |
| Rose (n petals) | r = a cos(nθ), n odd |
| Lemniscate | r² = a² cos(2θ) |
Key terms
- Polar coordinates
- A point (r, θ) where r is radial distance and θ is the angle from the positive x-axis.
- Pole
- The origin (0, 0) in the polar coordinate system.
- Cardioid
- Heart-shaped polar curve given by r = a(1 ± cos θ) or r = a(1 ± sin θ).
- Rose curve
- Petal-shaped curve r = a cos(nθ) with n petals (odd n) or 2n petals (even n).
Practice questions
Convert the polar point (4, π/3) to Cartesian coordinates.
- (2, 2√3)
- (4, 4)
- (2√3, 2)
- (√3, 1)
Show answer
x = 4 cos(π/3) = 4·(1/2) = 2; y = 4 sin(π/3) = 4·(√3/2) = 2√3.
Convert the Cartesian point (3, 3) to polar coordinates (r, θ) with r > 0 and 0 ≤ θ < 2π.
Show answer
r = √(9+9) = 3√2; θ = arctan(3/3) = arctan(1) = π/4.
How many petals does the rose curve r = 4 cos(3θ) have?
- 3
- 4
- 6
- 8
Show answer
For r = a cos(nθ), odd n gives n petals. Here n = 3, so 3 petals.
The polar equation r = 5 represents a circle of radius 5 centered at the origin.
Show answer
r = constant means all points are the same distance from the pole — a circle.
Convert the equation x² + y² = 16 to polar form.
Show answer
Since r² = x² + y², we have r² = 16, so r = 4.
Fun facts
- Polar coordinates were introduced by Jakob Bernoulli in 1691, though Isaac Newton used similar ideas slightly earlier.
- The Archimedean spiral r = aθ traces out an expanding coil — the spacing between each loop is constant, like a rolled-up ruler.
Get a personalized course on Polar Coordinates built around your child.