Polar to Cartesian Converter

How it works

Polar coordinates locate a point in the plane by its distance r from the origin and the angle θ its position makes with the positive x-axis, measured counter-clockwise. Going from polar to Cartesian is two multiplications: x = r cos θ and y = r sin θ. Going the other way, r is the hypotenuse √(x² + y²) and θ comes from the inverse tangent of y/x.

A plain arctangent cannot tell (3, 4) from (−3, −4), because both have y/x = 4/3. The calculator uses the two-argument atan2(y, x), which looks at the signs of x and y separately and so returns the correct quadrant: 53.13° for (3, 4) and −126.87° (equivalently 233.13°) for (−3, −4). The angle is reported both in the −180° to 180° range that atan2 gives and in the 0° to 360° range.

In three dimensions, cylindrical coordinates keep polar (ρ, θ) in the xy-plane and add the height z unchanged. Spherical coordinates use the straight-line distance r from the origin, the same azimuth θ, and the polar angle φ measured down from the positive z-axis, so x = r sin φ cos θ, y = r sin φ sin θ and z = r cos φ. The calculator reports every system for whichever point you enter; a 2D point is treated as lying in the plane z = 0, where φ = 90°.

Formula

Polar → Cartesian:      x = r cos θ          y = r sin θ
Cartesian → polar:      r = √(x² + y²)       θ = atan2(y, x)
Cylindrical:            ρ = √(x² + y²)       θ = atan2(y, x)       z = z
Spherical → Cartesian:  x = r sin φ cos θ    y = r sin φ sin θ     z = r cos φ
Cartesian → spherical:  r = √(x² + y² + z²)  θ = atan2(y, x)       φ = acos(z ÷ r)

Example

The Cartesian point (3, 4) is r = √(9 + 16) = 5 from the origin at θ = atan2(4, 3) ≈ 53.13°. The point (−1, 1) is at r = √2 ≈ 1.4142 and θ = 135°, in the second quadrant, where a bare atan(y/x) would wrongly give −45°.

The polar point (2, 60°) converts to x = 2 cos 60° = 1 and y = 2 sin 60° = √3 ≈ 1.7321. The spherical point (2, 45°, 60°) converts to x = y = 2 sin 60° cos 45° ≈ 1.2247 and z = 2 cos 60° = 1.

Assumptions and limitations

• θ is measured counter-clockwise from the positive x-axis, the mathematical convention. Compass bearings run clockwise from north; use the bearing calculator for those.
• Spherical coordinates follow the mathematics (calculus-textbook) convention: θ azimuth from +x, φ polar angle from +z; physics/ISO 80000-2 swaps the letters (θ polar, φ azimuth), so check which your source uses.
• The radius must be zero or positive. A negative polar radius (r, θ) is the same point as (|r|, θ + 180°); enter it that way.
• Angles entered in degrees are converted with π/180; results are rounded to four decimals for display only.

Frequently asked questions