Polar Coordinates and Equations
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn the polar coordinate system, how to convert between polar and rectangular coordinates, and how to work with basic polar equations.
The Concept
Section titled “The Concept”Up until now, every point you’ve plotted has used rectangular (Cartesian) coordinates: go x units left or right, then y units up or down. That system works great for lines, parabolas, and most algebra. But some shapes, especially circles, spirals, and anything involving rotation, are awkward to describe with x and y.
Polar coordinates offer a different approach. Instead of “how far left/right and how far up/down,” you describe a point by:
- r: how far from the origin (the distance)
- θ (theta): what angle from the positive x-axis (the direction)
Think of it like giving directions with a compass. Instead of saying “go 3 blocks east and 4 blocks north,” you’d say “go 5 blocks at a bearing of 53 degrees.” Same destination, different description.
The diagram above shows the point (3, 4) in rectangular coordinates. In polar, that same point is (5, 53°): it’s 5 units from the origin at an angle of about 53° from the positive x-axis. The concentric circles show distances (r = 1, 2, 3, 4, 5) and the spoke lines show angles.
The connection between the two systems comes from right triangle trigonometry. If you draw a right triangle from the origin to the point, the horizontal leg is x, the vertical leg is y, and the hypotenuse is r. That gives us the conversion formulas.
Rectangular to polar:
(Always check the quadrant when finding θ. The arctangent function only gives values in Q1 and Q4, so you may need to add π for points in Q2 or Q3.)
Polar to rectangular:
These are just SOH CAH TOA applied to the triangle: cos θ = x/r and sin θ = y/r, rearranged.
Once you’re comfortable with the coordinate system, you can write equations in polar form. Some curves that are complicated in rectangular form become beautifully simple in polar:
- r = a is a circle centered at the origin with radius a
- r = a cos θ is a circle with diameter a along the x-axis
- r = a sin θ is a circle with diameter a along the y-axis
Worked Example
Section titled “Worked Example”Example 1: Rectangular to polar
Convert the point (3, 4) to polar coordinates:
Polar form: (5, 53.13°) or approximately (5, 0.927 radians).
Example 2: Polar to rectangular
Convert (r, θ) = (4, π/3) to rectangular:
Rectangular form: approximately (2, 3.464).
Example 3: Convert an equation
Convert the rectangular equation x² + y² = 16 to polar form.
We know that x² + y² = r² in polar coordinates, so we can substitute directly:
That’s a circle of radius 4 centered at the origin. Much simpler in polar form.
Famous Polar Curves
Section titled “Famous Polar Curves”Beyond circles, polar coordinates can describe some beautiful and useful curves. Here are four of the most common.
Cardioid
Section titled “Cardioid”A cardioid is a heart-shaped curve. The equation r = a(1 + cos θ) produces this shape. The name comes from the Greek word for “heart.” You can think of it as what happens when you trace a point on the edge of a circle rolling around another circle of the same size. Cardioids show up in microphone pickup patterns (cardioid microphones reject sound from behind) and antenna design.
Limacon (with inner loop)
Section titled “Limacon (with inner loop)”A limacon looks like a cardioid’s more dramatic cousin. The equation r = a + b cos θ produces different shapes depending on the ratio of a to b. When b is greater than a (as in r = 1 + 2cos θ), the curve has an inner loop that passes back through the origin. When a equals b, you get a cardioid. When a is greater than b, you get a dimpled or convex limacon. The word comes from the French for “snail.”
Rose Curve (3 petals)
Section titled “Rose Curve (3 petals)”Rose curves produce petal-shaped patterns. The equation r = cos(nθ) creates petals whose count depends on n. When n is odd, you get exactly n petals. So r = cos(3θ) produces 3 petals. These curves are purely a product of the polar coordinate system and have no simple rectangular equivalent. They appear in signal analysis and mathematical art.
Rose Curve (4 petals)
Section titled “Rose Curve (4 petals)”When n is even, you get 2n petals. So r = cos(2θ) produces 4 petals, not 2. This is one of those results that surprises people at first but makes sense once you trace through the values of θ. The negative values of r (when cos(2θ) is negative) create the extra petals on the opposite side.
These curves are not just mathematical curiosities. Cardioid patterns are used in directional microphones. Rose curves appear in signal analysis. And limacon shapes show up in cam design for mechanical engineering.
Lemniscate (Infinity Symbol)
Section titled “Lemniscate (Infinity Symbol)”A lemniscate is a figure-eight or infinity symbol shape. The equation r² = a² cos(2θ) produces this curve. The name comes from the Latin “lemniscatus” meaning “decorated with ribbons.” It was studied by Jacob Bernoulli in 1694. The lemniscate is the set of all points where the product of the distances to two fixed points (foci) is constant. It shows up in physics (magnetic field lines), optics, and is the basis for the infinity symbol (∞) used in mathematics.
Archimedean Spiral
Section titled “Archimedean Spiral”An Archimedean spiral has the simple equation r = aθ, where the distance from the origin increases at a constant rate as the angle increases. Named after Archimedes who studied it around 225 BCE, this spiral has evenly spaced turns. You see it everywhere in nature and engineering: the grooves on a vinyl record, the coils of a spring, the way a garden hose coils, and the arms of some galaxies. It’s also the shape traced by a point moving outward at constant speed while rotating at constant speed.
Logarithmic Spiral
Section titled “Logarithmic Spiral”A logarithmic spiral has the equation r = ae^(bθ), where the distance from the origin grows exponentially with the angle. Unlike the Archimedean spiral, the turns get wider and wider as you go outward. This spiral has a remarkable property: it looks the same at every scale (self-similar), and it crosses every radius line at the same angle. It appears throughout nature: nautilus shells, hurricane cloud bands, spiral galaxies, the flight path of a hawk approaching prey, and the arrangement of seeds in a sunflower. Jacob Bernoulli was so fascinated by it that he asked for one to be engraved on his tombstone with the words “Eadem mutata resurgo” (“Although changed, I arise the same”).
Real-World Application
Section titled “Real-World Application”Polar coordinates are used in:
- Radar and sonar systems (distance and bearing angle are naturally polar)
- Astronomy (planetary orbits are described in polar coordinates)
- Navigation (GPS bearings, compass headings)
- Electrical engineering (phasor diagrams for AC circuits)
- Computer graphics (rotations, spiral patterns)
- Physics (central force problems, orbital mechanics)
Example: radar systems naturally use polar coordinates. The screen shows distance from the center (r) and direction (θ), which is exactly a polar grid.