Every point on a plane has two equally valid addresses: its Cartesian coordinates (x, y) — how far to walk horizontally, then vertically from the origin — and its polar coordinates (r, θ) — how far to walk in a straight line, and in which direction. Switching between them is pure trigonometry, yet the conversions trip up students and engineers alike because of the four-quadrant atan2 subtlety and the radian/degree distinction. This tool handles all of it in real time, shows the full substituted working, and draws the resulting point on a live diagram.
The same ideas extend into three dimensions as spherical coordinates (r, θ, φ), used throughout physics, chemistry, computer graphics and geospatial work. The calculator covers both the 2D and 3D cases.
How it works
2D: Polar → Cartesian
A polar point (r, θ) sits at distance r from the origin, at angle θ measured anticlockwise from the positive x-axis. Project that onto the two axes using the definitions of cosine and sine:
x = r · cos(θ) y = r · sin(θ)
The result panel shows each substitution explicitly — for example, r = 5 and θ = 53.13° gives x = 5 · cos(53.13°) = 3 and y = 5 · sin(53.13°) = 4.
2D: Cartesian → Polar
The radius is the Euclidean distance from the origin (Pythagorean theorem):
r = √(x² + y²)
The angle uses the two-argument arctangent so the correct quadrant is always returned:
θ = atan2(y, x)
atan2 returns values in (−π, π] radians, which the tool converts to degrees when you have degrees selected. The colour badge below the result shows which quadrant the point falls in.
3D: Spherical → Cartesian
Following the physics (ISO 80000-2) convention where θ is the polar angle from the +z-axis and φ is the azimuthal angle from the +x-axis:
x = r · sin(θ) · cos(φ) y = r · sin(θ) · sin(φ) z = r · cos(θ)
The inverse recovers r = √(x²+y²+z²), θ = arccos(z/r), and φ = atan2(y, x).
Worked example
Point P = (3, 4) in Cartesian coordinates:
- r = √(3² + 4²) = √(9 + 16) = √25 = 5
- θ = atan2(4, 3) ≈ 53.13° (or ≈ 0.9273 rad)
- Quadrant: I (both coordinates positive)
Verify in the other direction: x = 5 · cos(53.13°) ≈ 3, y = 5 · sin(53.13°) ≈ 4. ✓
| Start | r or (x, y) | θ | Result |
|---|---|---|---|
| Polar → Cart | r = 1, θ = 0° | 0° | (1, 0) |
| Polar → Cart | r = 1, θ = 90° | 90° | (0, 1) |
| Polar → Cart | r = 5, θ = 53.13° | 53.13° | (3, 4) |
| Cart → Polar | (1, 1) | — | r = √2 ≈ 1.4142, θ = 45° |
| Cart → Polar | (0, −3) | — | r = 3, θ = −90° |
Formula note
The conversion relies on the unit circle definitions of sine and cosine: for a point on a circle of radius r, the horizontal displacement is r·cos(θ) and the vertical is r·sin(θ). These are exact when r = 1 (the unit circle) and scale linearly for any r. The 3D formulas follow by first collapsing the point onto the xy-plane (distance r·sin(θ) from the z-axis) and then projecting that shadow onto x and y using the azimuthal angle φ. Every step is standard trigonometry — no approximations are involved.
Everything is calculated locally in your browser — no values are sent to any server.