Convert Cartesian (x, y) coordinates to polar (r, θ) and back, with batch mode, coordinate plot, and conversion reference table.
Converting between Cartesian (rectangular) and polar coordinate systems is a fundamental operation in mathematics, physics, and engineering. While Cartesian coordinates specify a point using horizontal (x) and vertical (y) distances from the origin, polar coordinates use a radius (r) — the distance from the origin — and an angle (θ) measured from the positive x-axis.
This converter handles both directions: Cartesian → Polar using r = √(x² + y²) and θ = atan2(y, x), and Polar → Cartesian using x = r·cos(θ) and y = r·sin(θ). The atan2 function correctly handles all four quadrants, unlike the basic arctan. A batch mode converts multiple points simultaneously, and a visual coordinate plot shows the point's location.
Polar coordinates are especially useful for problems with circular symmetry: orbits, spirals, wave patterns, antenna radiation patterns, and complex number operations. Understanding both systems and converting fluently between them is essential in calculus (double integrals), physics (orbital mechanics), signal processing, and computer graphics (rotation transformations).
Converting coordinates by hand requires evaluating square roots and inverse tangent functions while correctly identifying the quadrant — the basic arctan function only covers half the plane, making atan2 essential. Batch conversions of multiple points compound these issues. This converter handles all four quadrants correctly using atan2, supports both degree and radian modes, converts multiple points at once in batch mode, and plots each result visually. It is ideal for students checking trigonometry homework, engineers decomposing force vectors, and anyone working with polar curves or complex number arithmetic.
r = √(x² + y²) θ = atan2(y, x) x = r · cos(θ) y = r · sin(θ)
Result: r = 5, θ ≈ 53.13°
r = √(9 + 16) = 5. θ = atan2(4, 3) ≈ 53.13°. This is the classic 3-4-5 right triangle.
Cartesian coordinates excel at describing rectangular geometry — grids, boxes, linear motion. Polar coordinates excel at circular geometry — orbits, spirals, rotations. The choice depends on the problem's symmetry. Integrating over a circular region is painful in Cartesian but natural in polar (r dr dθ). Conversely, describing a rectangle is trivial in Cartesian but awkward in polar. Fluency in both systems and quick conversion between them is a core mathematical skill.
The standard arctan(y/x) function cannot distinguish between quadrants I and III (both give positive y/x) or quadrants II and IV (both give negative y/x). The atan2(y, x) function solves this by examining the signs of both x and y independently, returning angles in the full range (−180°, 180°]. This is why every programming language and scientific calculator provides atan2 — it is the correct function for coordinate conversion.
Every complex number z = x + iy can be written in polar form z = r·e^(iθ), where r = |z| = √(x²+y²) and θ = arg(z) = atan2(y,x). Multiplication in polar form is elegant: z₁·z₂ = r₁r₂·e^(i(θ₁+θ₂)) — multiply magnitudes, add angles. This is why converting to polar form simplifies complex arithmetic, roots of unity calculations, and Fourier analysis. De Moivre's theorem [r·e^(iθ)]ⁿ = rⁿ·e^(inθ) is another powerful result that relies on polar representation.
Cartesian uses (x, y) perpendicular distances; polar uses (r, θ) distance and angle from the origin. Use this as a practical reminder before finalizing the result.
atan2(y, x) returns the correct angle in all four quadrants (−180° to 180°), while atan(y/x) only covers −90° to 90°. Keep this note short and outcome-focused for reuse.
Conventionally r ≥ 0, but some contexts allow negative r, meaning the point is in the opposite direction: (−r, θ) = (r, θ + 180°). Apply this check where your workflow is most sensitive.
Multiply by 180/π. Conversely, degrees to radians: multiply by π/180.
For problems with circular symmetry — orbits, spirals, rotation, complex multiplication, and polar integrals. Use this checkpoint when values look unexpected.
In 3D, spherical coordinates (r, θ, φ) or cylindrical coordinates (r, θ, z) extend the polar concept. Validate assumptions before taking action on this output.