Polar Coordinates Calculator

Polar coordinates represent a point in the plane using a distance r from the origin and an angle θ from the positive x-axis. They are written as (r, θ) and are especially useful for circular and spiral patterns.

Use r = √(x² + y²) for the distance and θ = atan2(y, x) for the angle. The atan2 function correctly handles all four quadrants, unlike a simple arctan(y/x) which only works in Quadrants I and IV.

atan2(y, x) considers the signs of both x and y to determine the correct quadrant. A simple arctan(y/x) cannot distinguish between Quadrants I and III or II and IV, leading to angles off by 180°.

Conventionally r ≥ 0. A negative r value means you go in the opposite direction of the angle θ (equivalent to adding 180° to θ and using positive r). This calculator uses r ≥ 0.

Use x = r·cos(θ) and y = r·sin(θ). Make sure θ is in the correct unit (degrees or radians) matching your calculator settings.

The angle θ is typically expressed in the range (-180°, 180°] or [0°, 360°). This calculator outputs θ in (-180°, 180°] and also provides the equivalent [0°, 360°) value.

Polar coordinates are used in navigation (bearing and distance), physics (circular motion, central forces), engineering (antenna patterns), and mathematics (spirals, roses, cardioids).

A complex number z = x + iy can be written in polar form as z = r(cos θ + i sin θ) = r·e^(iθ), where r = |z| is the modulus and θ = arg(z) is the argument.

Add 360° to a negative angle to get the equivalent positive angle. For example, -30° + 360° = 330°.

Cartesian coordinates use horizontal (x) and vertical (y) distances from the origin. Polar coordinates use radial distance (r) and angle (θ). Polar is better for circular patterns.

Points on the positive x-axis have θ = 0°, positive y-axis θ = 90°, negative x-axis θ = 180°, and negative y-axis θ = 270° (or -90°).

Yes, the origin is represented as (0, θ) for any angle θ, since r = 0 means the point is at the origin regardless of the angle.

Start at the origin, measure the angle θ from the positive x-axis, then move r units in that direction to locate the point.

Circles, spirals (Archimedean, logarithmic), rose curves, cardioids, limaçons, and lemniscates are all naturally expressed in polar coordinates.

Substitute r = √(x² + y²), x = r·cos(θ), and y = r·sin(θ) into the polar equation and simplify.

atan2(y, x) is a two-argument arctangent function that returns the correct angle for all four quadrants by considering the signs of both x and y.

Multiply radians by 180/π to convert to degrees, or multiply degrees by π/180 to convert to radians. This calculator provides both.

A complex number z = x + iy can be written as z = r(cos θ + i sin θ) = r·e^(iθ), where r is the modulus and θ is the argument.

Polar coordinates simplify problems involving circular motion, central forces (gravity, electrostatics), and rotational dynamics.

Spherical coordinates extend polar coordinates to 3D by adding a third coordinate (φ or ρ) for the angle from the z-axis or distance from the origin.

Yes, (r, θ) with r < 0 is equivalent to (-r, θ + 180°). This calculator normalizes to r ≥ 0.

Use the law of cosines: d = √(r₁² + r₂² - 2r₁r₂cos(θ₂ - θ₁)).

The polar coordinate system was developed by Gregorio Fontana in the 18th century, building on earlier work by Isaac Newton and Jacob Bernoulli.

If x = f(t) and y = g(t), substitute into r = √(x² + y²) and θ = atan2(y, x) to get polar equations in terms of the parameter t.

The slope of the tangent line to a polar curve r = f(θ) is given by dy/dx = (dr/dθ·sin θ + r·cos θ)/(dr/dθ·cos θ - r·sin θ).

The area is A = ½∫[r(θ)]² dθ over the appropriate interval for θ.

The Polar Coordinates Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Polar Coordinates Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.