Polar coordinates
Polar coordinates describe a point by how far it is from the origin, r, and the angle θ it makes with the positive x-axis. Curves that are awkward in x and y, such as spirals, roses and cardioids, become one-line equations in r and θ.
Distance and direction
A point with polar coordinates (r, θ) is found by facing along the positive x-axis, turning anticlockwise through θ, and walking r units. The angle is usually in radians. (2, π/2) is two units straight up, the Cartesian point (0, 2). The same point has many polar names: adding 2π to θ, or using (−r, θ + π), gets you to the same place.
Converting between polar and Cartesian
Polar to Cartesian: x = r cos θ, y = r sin θ.
Cartesian to polar: r = √(x² + y²), and θ is the angle with tan θ = y/x, chosen in the right quadrant. The two-argument function atan2(y, x) does that quadrant bookkeeping for you.
Example: (−1, √3) has r = √(1 + 3) = 2. y/x = −√3, and the point is in quadrant II, so θ = 2π/3 rather than −π/3.
Polar equations
A polar equation gives r in terms of θ. The simplest ones are r = c, a circle of radius c, and θ = c, a straight line through the origin. Converting an equation uses the same substitutions: r = 2cos θ becomes r² = 2r cos θ, so x² + y² = 2x, a circle of radius 1 centred at (1, 0).
In the calculator, type r = 2cos(theta). θ sweeps from 0 to 12π by default, which closes almost any polar curve.
Negative r
When an equation gives a negative r, the point is plotted in the opposite direction: (−r, θ) is the same as (r, θ + π). That is why the limaçon r = 1 + 2cos θ has an inner loop: for θ near π, 1 + 2cos θ is negative and the curve swings back through the origin to the other side.
The common polar curves
| Curve | Equation |
|---|---|
| Circle through the origin | r = 2a cos θ |
| Cardioid | r = a(1 + cos θ) |
| Limaçon | r = a + b cos θ |
| Rose with n or 2n petals | r = a cos(nθ) |
| Archimedean spiral | r = aθ |
| Lemniscate | r² = a² cos 2θ |
Area in polar coordinates
The area swept out between θ = α and θ = β is ½ ∫ r² dθ. For the cardioid r = 1 + cos θ over a full turn that is ½ ∫ (1 + cos θ)² dθ from 0 to 2π = 3π/2. In the calculator, int((1 + cos(t))^2, t, 0, 2pi)/2 returns 4.71238898038, which is 3π/2.
Practice
A new problem every time. Answer in any form you like: fractions, decimals, √ and π all work, and equivalent equations count as right.
Convert the polar point (r, θ) = (3, π/6) to Cartesian coordinates (x, y).
Show the answer
Answer: (3√3/2, 3/2)
- x = r cos θ and y = r sin θ.
- cos(π/6) = √3/2 and sin(π/6) = 1/2.
- x = 3 · √3/2 = 3√3/2, y = 3 · 1/2 = 3/2.
- The point is (3√3/2, 3/2). Decimals to two places are fine too.
Frequently asked questions
How do you convert polar to Cartesian coordinates?
How do you convert Cartesian to polar coordinates?
What does a negative r mean?
How do I graph a polar equation?
Keep going
Every figure on this page opens in ihateDesmos, where you can drag sliders and watch the graph respond.