How this instrument works
Rectangular (Cartesian) coordinates describe a point by how far it sits along two perpendicular axes, x and y. Polar coordinates describe that identical point instead by its distance r from the origin and its angle θ from a reference direction. Converting from rectangular to polar applies the Pythagorean theorem for the distance, r = √(x²+y²), and the two-argument arctangent for the angle, θ = atan2(y, x).
atan2 matters here specifically because a plain single-argument arctangent, atan(y⁄x), can't tell which quadrant a point sits in — it only sees the ratio between y and x, and that ratio looks identical for a point and its exact opposite. atan2 reads both coordinates' individual signs and places the resulting angle correctly, automatically, with no separate quadrant check needed afterward.
This is exactly the inverse of this site's companion Polar to Rectangular page. Together, the pair let a position move freely between the two coordinate systems — plotted map coordinates, game-engine positions, and sensor output arriving in plain (x, y) form can all be converted to distance-and-bearing when a downstream system, like a radar display or a navigation instrument, needs that form instead.
- Enter the point's horizontal position into the x field.
- Enter its vertical position into the y field.
- Read r and θ: the sheet computes the distance and angle simultaneously.
Worked example — the point (3, 4)
The rectangular point (3, 4) converts to polar form as distance 5, angle ≈53.13° — a direct Pythagorean-theorem distance and an atan2-derived angle, the same 3-4-5 relationship showing up here as a coordinate conversion rather than a triangle's own sides.
The point (10, 0) converts to distance 10, angle 0° — sitting exactly on the reference direction. The point (0, 5) converts to distance 5, angle 90° instead — a quarter turn straight up the perpendicular axis, with no component along the original reference direction remaining.
Questions
How do you convert rectangular coordinates to polar?
Use the Pythagorean theorem for the distance, r = √(x²+y²), and the two-argument arctangent for the angle, θ = atan2(y, x). Together these give the equivalent distance-and-angle description of a plain (x, y) point.
Why use atan2 instead of a plain arctangent?
A plain arctan(y⁄x) only sees the ratio of the two coordinates, which is identical for a point and its exact opposite — (3,4) and (−3,−4) share the same ratio despite sitting in opposite directions. atan2 reads both signs and places the angle in the correct quadrant automatically.
Is this the inverse of the Polar to Rectangular page?
Yes — the two pages undo each other exactly. Converting a point to polar form and then back to rectangular with this site's companion page returns the identical original coordinates.
When would I need this conversion?
Whenever a position tracked in plain (x, y) form — a plotted map point, a game-engine object, sensor output — needs feeding into a system that expects distance-and-bearing instead, such as a radar display, a robotic arm's reach-and-swing control, or a navigation instrument.
What is the angle of the origin point (0, 0)?
Undefined — a point with zero distance from the origin has no meaningful direction, since it doesn't sit anywhere away from that reference point at all. This calculator flags a (0, 0) input rather than returning an arbitrary angle.