Why atan2 belongs in a polar converter
Hand-typed arctan(y ÷ x) breaks the moment x crosses zero or the pair sits left of the y-axis. Browsers expose Math.atan2(y, x), which returns a signed angle consistent with quadrants. This page uses atan2 for every rectangular update so θ tracks your point around the full plane.
Short version: you keep thinking in slopes, the tool keeps thinking in half-turns and full turns.
Heads-up: at the origin, r is zero and θ is undefined. The plot still marks (0, 0), but any angle label is meaningless there. Courses still ask for θ anyway, so know the convention your grader expects.
Three places polar form earns its keep
Antenna and radar sketches often start as magnitude-plus-bearing. You drop to x and y only when Cartesian math is easier. Robotics arms use polar-like segments chained together. Complex numbers on an Argand diagram are basically this same (r, θ) story, which is why our complex number calculator pairs naturally with this converter.
- Physics labs: convert a 2.4 m displacement at 37° from east into x and y components for vector sums.
- Graphics: rotate a sprite by θ while keeping distance r from a hinge.
- Curves: switch representations before feeding a curve into a parametric plotter or comparing against a Cartesian function grapher.
What runs in your browser
Each keystroke recomputes r, θ, x, and y locally. Nothing is posted to a server. The canvas redraw uses your current numbers, optional grid, radius segment, and angle arc. Floating-point rounding shows up past the fourth decimal, same as any JavaScript math.
If you need strict symbolic π multiples (π/4, 2π/3) printed as symbols, pencil-and-paper still wins. Here you get decimal values suited to homework checks and quick sanity tests.
What this page refuses to guess for you
We do not pick a canonical θ when r = 0. We do not normalize θ to (−π, π] versus [0, 2π) unless you steer the input yourself. Textbooks disagree on those details, so the UI stays literal: degrees shown above zero follow the usual 0°-360° wrap from atan2.
For distance-only questions, the distance calculator or Pythagorean tool skips angle noise entirely. For 3-D work, move on to the vector calculator instead of forcing polar tricks.
Degrees, radians, and the east reference
θ is measured counterclockwise from the positive x-axis (east in the default sketch). Radians option changes only the θ input box and the numeric radian readout in the chips above. The arc overlay label stays in degrees for quick visual reading.
- Positive x, positive y
- Quadrant I. θ matches what intuition expects from a slope triangle.
- Negative x
- Quadrant II or III. atan2 adds the half-turn logic so you do not paste the wrong branch.
- Negative r
- Not accepted here. Enter r ≥ 0. Some courses allow “negative radius” with θ + π, we skip that path to avoid silent flips.
Toolexe ships small, focused math utilities. Polar conversion is standard analytic geometry: r = √(x² + y²), x = r cos θ, y = r sin θ, with θ from atan2 when starting from x and y. Spot something off? Re-check your angle unit toggle first.
