What is the formula for Cartesian to Polar conversion?

The standard formulas are r = √(x² + y²) and θ = atan2(y, x). atan2 is the four-quadrant inverse tangent: it returns an angle in (−π, π] and handles x = 0 correctly. The calculator normalises this to [0, 360°) or [0, 2π) so the angle is always positive.

What is the formula for Polar to Cartesian conversion?

x = r · cos(θ) and y = r · sin(θ). These come directly from the definition of the unit circle. If r is negative (physically unusual but sometimes seen in signed-radius conventions), the point mirrors through the origin.

What is the difference between the math and geographic angle conventions?

In the math convention θ is measured from the positive x-axis, counter-clockwise. In the geographic (navigation/surveying) convention the angle is a bearing measured from North (the positive y-axis), clockwise. To convert: bearing = 90° − math-angle (mod 360°). The calculator handles both so you never need to do that manually.

Why does the calculator use atan2 instead of arctan(y/x)?

arctan(y/x) is ambiguous: points (1, 1) and (−1, −1) both give arctan(1) = 45°, yet they lie in opposite quadrants. atan2(y, x) accepts the two values separately and uses the signs of both to place the angle in the correct quadrant, giving a unique result for every non-origin point.

What happens when x = 0 or y = 0?

The calculator handles all axis cases. When x = 0 and y > 0 the angle is 90° (π/2 rad). When x = 0 and y < 0 it is 270° (3π/2 rad). When y = 0 and x > 0 the angle is 0°. At the origin (x = 0, y = 0) the radius is 0 and the angle is mathematically undefined — the calculator returns 0 for θ in that case.

Can I use this for 3D spherical coordinates?

This tool covers the 2D case (x, y) ↔ (r, θ). For 3D spherical coordinates (x, y, z) ↔ (ρ, θ, φ) you additionally need ρ = √(x² + y² + z²), polar angle φ = arccos(z/ρ), and azimuthal angle θ = atan2(y, x). The 2D radial distance here equals the 3D quantity √(x² + y²), which is the projection onto the xy plane.

Cartesian to Polar Calculator

Converting between Cartesian coordinates (x, y) and polar coordinates (r, θ) is a fundamental operation in mathematics, physics, engineering, navigation, and computer graphics. A Cartesian point describes a location by its horizontal and vertical offsets from an origin; a polar point describes the same location by a radial distance r and an angle θ from a reference direction. Both representations are exact and lossless — every Cartesian point has a unique polar counterpart (except the origin, where θ is undefined), and vice versa.

The standard formulas

Cartesian to Polar:

r = √(x² + y²) — the Euclidean distance from the origin, always ≥ 0.
θ = atan2(y, x) — the four-quadrant inverse tangent, returning a value in (−π, π].

The calculator normalises θ to the range [0°, 360°) (or [0, 2π) in radians) so the angle is always a positive, unambiguous value. The signed form from atan2 is also shown for reference.

Polar to Cartesian:

x = r · cos(θ)
y = r · sin(θ)

These follow directly from the unit-circle definitions of cosine and sine.

Angle conventions explained

The math convention (also called the standard mathematical convention) measures θ counter-clockwise from the positive x-axis. This is what textbooks, physics equations like v = r·ω, and most programming libraries use by default.

The geographic convention measures θ as a bearing clockwise from North (the positive y-axis). Bearing = atan2(x, y), normalised to [0°, 360°). This is used in navigation, surveying, mapping APIs, and aviation. The calculator supports both so you can work directly in your domain without manual remapping.

Worked example

A point at (x = 3, y = 4) in the math convention, degrees:

r = √(3² + 4²) = √(9 + 16) = √25 = 5
θ = atan2(4, 3) ≈ 53.130°

So the polar form is (r = 5, θ ≈ 53.13°).

Reversing: start from (r = 5, θ = 53.13°):

x = 5 · cos(53.13°) ≈ 5 × 0.6 = 3
y = 5 · sin(53.13°) ≈ 5 × 0.8 = 4

Round-trip recovers the original point exactly (within floating-point precision).

x	y	r	θ (math, deg)	Quadrant
1	0	1	0°	+x axis
0	1	1	90°	+y axis
−1	0	1	180°	−x axis
0	−1	1	270°	−y axis
3	4	5	53.130°	I
−3	4	5	126.870°	II
−3	−4	5	233.130°	III
3	−4	5	306.870°	IV

Where polar coordinates are used

Physics: Circular motion equations (centripetal force F = mv²/r, angular velocity ω = dθ/dt), planetary orbits (Kepler’s laws are elegant in polar form), wave optics, and electromagnetic field problems with radial symmetry all use polar or cylindrical coordinates naturally.

Engineering: Antenna radiation patterns, signal processing (complex numbers in polar form: z = r·e^(iθ)), control systems (Bode plots use the polar form of a transfer function), and stress tensors in circular cross-sections.

Navigation: Aircraft and ship bearings are geographic angles. Radar returns a (range, bearing) pair in polar form. GPS receivers convert ECEF Cartesian coordinates to latitude/longitude which is effectively spherical polar.

Computer graphics: Path-based drawing APIs (SVG arc commands, canvas arc), rotation transforms, particle systems, and procedural animation often work in polar coordinates and convert to screen-space Cartesian only at the rendering step.

Mathematics: Curves like the Archimedean spiral (r = aθ), rose curves (r = cos(nθ)), cardioids (r = 1 + cos θ), and lemniscates are far simpler in polar form than Cartesian. Integration over circular or annular regions is also dramatically easier in polar form using the Jacobian r dr dθ.