Polar Coordinates — Definition, Formula & Examples

Polar Coordinates

A way to describe the location of a point on a plane. A point is given coordinates (r, θ). r is the distance from the point to the origin. θ is the angle measured counterclockwise from the polar axis to the segment connecting the point to the origin.

Note: With polar coordinates a given point has many possible representations. θ has many possible values depending on which coterminal angle is chosen, and r can be positive or negative.

Polar coordinate diagram showing point (r, θ), with r as length from origin and θ as angle from horizontal polar axis.

Key Formula

x = r\cos\theta, \quad y = r\sin\theta \qquad \text{and} \qquad r = \sqrt{x^2 + y^2}, \quad \theta = \arctan\!\left(\frac{y}{x}\right)

Where:

$r$ = The distance from the point to the origin (can be negative in polar coordinates)
$\theta$ = The angle measured counterclockwise from the polar axis (positive x-axis) to the line segment connecting the origin to the point
$x$ = The horizontal (rectangular) coordinate of the point
$y$ = The vertical (rectangular) coordinate of the point

Worked Example

Problem: Convert the rectangular (Cartesian) point (3, 3) to polar coordinates.

Step 1: Find r using the distance formula from the origin.

r = \sqrt{x^2 + y^2} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}

Step 2: Find θ using the arctangent formula. Since both x and y are positive, the point is in Quadrant I.

\theta = \arctan\!\left(\frac{y}{x}\right) = \arctan\!\left(\frac{3}{3}\right) = \arctan(1) = \frac{\pi}{4}

Step 3: Write the polar coordinates as an ordered pair (r, θ).

\left(3\sqrt{2},\; \frac{\pi}{4}\right)

Answer: The polar coordinates of (3, 3) are

(3\sqrt{2},\; \frac{\pi}{4})

Another Example

This example goes in the reverse direction—from polar to rectangular—showing how to apply the cosine and sine conversion formulas directly.

Problem: Convert the polar coordinates

(5, \frac{2\pi}{3})

to rectangular coordinates.

Step 1: Find x using the conversion formula x = r cos θ.

x = 5\cos\!\left(\frac{2\pi}{3}\right) = 5 \cdot \left(-\frac{1}{2}\right) = -\frac{5}{2}

Step 2: Find y using the conversion formula y = r sin θ.

y = 5\sin\!\left(\frac{2\pi}{3}\right) = 5 \cdot \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2}

Step 3: Write the rectangular coordinates.

\left(-\frac{5}{2},\; \frac{5\sqrt{3}}{2}\right)

Answer: The rectangular coordinates are

\left(-\frac{5}{2},\; \frac{5\sqrt{3}}{2}\right)

Frequently Asked Questions

What is the difference between polar and rectangular coordinates?

Rectangular (Cartesian) coordinates locate a point using horizontal and vertical distances (x, y) from the origin. Polar coordinates locate the same point using a distance r from the origin and an angle θ from the positive x-axis. You can convert between the two systems using the formulas x = r cos θ, y = r sin θ, and r = √(x² + y²).

Why can a point have more than one set of polar coordinates?

Because angles are periodic, adding any multiple of 2π to θ gives the same direction. For example, (3, π/4) and (3, 9π/4) represent the same point. Additionally, using a negative r with an angle shifted by π also reaches the same location: (−3, 5π/4) is the same point as (3, π/4). This means every point has infinitely many polar representations.

When should you use polar coordinates instead of rectangular?

Polar coordinates are especially useful when a problem involves circles, spirals, or rotational symmetry. Equations like r = 5 (a circle of radius 5) are much simpler in polar form than their rectangular equivalent x² + y² = 25. They also simplify many calculus problems involving curves like cardioids, limaçons, and rose curves.

Polar Coordinates vs. Rectangular (Cartesian) Coordinates

	Polar Coordinates	Rectangular (Cartesian) Coordinates
Definition	Locates a point using distance r and angle θ	Locates a point using horizontal distance x and vertical distance y
Notation	(r, θ)	(x, y)
Uniqueness	A point has infinitely many representations	Each point has exactly one representation
Best for	Circles, spirals, rotational symmetry	Lines, parabolas, general-purpose graphing
Circle equation	r = a (very simple)	x² + y² = a² (more complex)

Why It Matters

Polar coordinates appear in precalculus when you study conic sections and again in calculus when computing areas enclosed by curves like cardioids and rose curves. They are essential in physics and engineering for describing circular motion, wave patterns, and fields with radial symmetry. Understanding how to convert between polar and rectangular forms is a core skill on the AP Calculus BC exam and in multivariable calculus.

Common Mistakes

Mistake: Using arctan(y/x) without checking the quadrant of the point.

Correction: The basic arctan function only returns values in (−π/2, π/2), which covers Quadrants I and IV. If the point is in Quadrant II or III, you must add π to the arctan result (or use the atan2 function) to get the correct angle.

Mistake: Assuming polar coordinates are unique, just like rectangular coordinates.

Correction: Unlike (x, y), a single point has infinitely many polar representations. For instance, (r, θ), (r, θ + 2π), and (−r, θ + π) all describe the same point. Be careful when comparing two polar coordinate pairs—they may look different but represent the same location.

Related Terms

Polar-Rectangular Conversion Formulas — Formulas to convert between the two systems
Area Using Polar Coordinates — Calculus formula for area of polar curves
Polar Axis — The reference direction for measuring θ
Coterminal Angles — Explains why θ is not unique
Parametric Equations — Another way to represent curves in a plane
Polar Derivative Formulas — Calculus derivatives for polar curves
Origin — The reference point where r = 0
Point — The geometric object located by coordinates