Polar to Cartesian Converter

Reviewed by Hamza Gujjar, Algorithm Architect & Software Developer Last updated: August 19, 2026

📚 Confused about how this is calculated? Read the full Two Languages for the Same Point: Coordinate Systems →

Two Ways to Name the Same Point

Every point on a plane can be located two different ways: Cartesian coordinates give its horizontal and vertical distance from the origin, while polar coordinates give its straight-line distance and the angle from a reference direction. Both describe the exact same point — converting between them is a matter of trigonometry, not measurement.

The Formulas

Polar to Cartesian: x = r×cos(θ), y = r×sin(θ)
Cartesian to Polar: r = √(x²+y²), θ = atan2(y, x)

Worked Examples

Conversions in both directions
Direction	Input	Output
Polar → Cartesian	r = 10, θ = 30°	x = 8.6603, y = 5
Cartesian → Polar	x = 3, y = 4	r = 5, θ = 53.1301°

The second example is the familiar 3-4-5 right triangle, so r comes out to a clean 5.

Where This Matters

Radar and sonar tracking — targets are naturally located by range and bearing (polar), then converted to x/y for mapping and plotting.
Robotics arm control — a robotic arm often reasons in polar terms (reach and angle) while the workspace is defined in Cartesian coordinates.
Complex numbers and signal processing — polar form (magnitude and phase) and rectangular form (real and imaginary parts) are the same conversion problem in a different notation.

How to Use This Calculator

Choose the direction: Polar to Cartesian or Cartesian to Polar.
For polar to Cartesian, enter Radius (r) and Angle (theta), then choose Degrees or Radians.
For Cartesian to polar, enter x and y, and choose whether the angle result should be in Degrees or Radians.
Select Calculate to get the converted coordinates.

Related Calculations

See these same cosine and sine relationships applied at radius 1 in the Unit Circle Calculator.

Principles of Coordinate Geometry and Spatial Transformations

A polar to Cartesian coordinate converter transforms two-dimensional and three-dimensional spatial coordinates between the 2D Cartesian (x, y), 2D Polar (r, θ), 3D Cylindrical (r, θ, z), and 3D Spherical (ρ, θ, φ) coordinate reference systems. In robotic kinematics, CNC machining, radar tracking, and vector calculus, converting between coordinate systems simplifies trajectory calculations and multi-variable integration.

The Fundamental 2D Transformation Equations

Cartesian to Polar (x, y → r, θ):
Radial Distance: r = √( x² + y² )
Azimuthal Angle: θ = atan2( y, x ) (in radians or degrees)

Polar to Cartesian (r, θ → x, y):
x = r × cos(θ)
y = r × sin(θ)

Four-Quadrant Arctangent Function (atan2)

Unlike standard arctangent (which returns values restricted to -π/2 ≤ θ ≤ π/2), the atan2(y, x) function resolves quadrant ambiguities across all 360 degrees:

Quadrant I (x > 0, y > 0): θ = arctan(y/x) (0° to 90°).
Quadrant II (x < 0, y > 0): θ = arctan(y/x) + 180° (90° to 180°).
Quadrant III (x < 0, y < 0): θ = arctan(y/x) - 180° (-180° to -90° or 180° to 270°).
Quadrant IV (x > 0, y < 0): θ = arctan(y/x) (-90° to 0° or 270° to 360°).

3D Spherical Coordinate Transformation Equations

Radius: ρ = √( x² + y² + z² ) | Polar Inclination: φ = arccos( z / ρ ) | Azimuth: θ = atan2( y, x )
x = ρ × sin(φ) × cos(θ) | y = ρ × sin(φ) × sin(θ) | z = ρ × cos(φ)

Step-by-Step Worked Calculation Example

Example: Converting Cartesian Point P(-3.0, 4.0) into 2D Polar Coordinates

Problem: Convert the 2D Cartesian coordinates x = -3.0 and y = +4.0 into polar coordinates (r, θ) in both radians and degrees.

Step 1: Calculate radial distance r:

r = √[ (-3.0)² + (4.0)² ] = √[ 9.0 + 16.0 ] = √25.0 = 5.000 Units

Step 2: Calculate angle θ using Quadrant II atan2 logic:

Base Angle = arctan( |4.0| / |-3.0| ) = arctan( 1.3333 ) = 0.9273 radians (53.13°)

θ = 180.0° - 53.13° = 126.87° (2.2143 Radians)

Conclusion: Cartesian (-3.0, 4.0) transforms into Polar coordinates (r = 5.0, θ = 126.87°).

Vector Calculus and Multi-Variable Integration Transformations

In advanced calculus and electromagnetic physics, converting surface and volume integrals from Cartesian coordinates into polar, cylindrical, or spherical coordinates simplifies boundary equations. The infinitesimal area and volume differential elements scale by the Jacobian Determinant:

2D Polar Differential Area Element: dA = dx dy = r × dr dθ
3D Cylindrical Differential Volume: dV = dx dy dz = r × dr dθ dz
3D Spherical Differential Volume: dV = dx dy dz = ρ² × sin(φ) × dρ dφ dθ

3D Cylindrical Coordinate System (r, θ, z)

Cylindrical coordinates extend 2D polar geometry along a perpendicular vertical z-axis:

Cartesian from Cylindrical: x = r × cos(θ), y = r × sin(θ), z = z
Cylindrical from Cartesian: r = √(x² + y²), θ = atan2(y, x), z = z
Engineering Application: Ideal for analyzing rotating machinery shafts, fluid flow in circular pipes, and magnetic fields inside solenoids.

2D Homogeneous Coordinate Transformation Matrices

In computer graphics and robotic kinematics, rotating a point vector by an angle θ counter-clockwise about the origin is performed via matrix multiplication:

[ x' ] = [ cos(θ) -sin(θ) ] × [ x ]
[ y' ] = [ sin(θ) cos(θ) ] × [ y ]

Vector Operations in Polar and Spherical Forms

In vector physics and classical electromagnetic field theory, vector dot and cross products are formulated using coordinate components:

2D Vector Dot Product (A · B):
A · B = |A| × |B| × cos(Δθ) = ( x₁ × x₂ ) + ( y₁ × y₂ )
Measures parallel directional alignment between spatial force vectors.
Vector Cross Product (A × B):
|A × B| = |A| × |B| × sin(Δθ)
Calculates the perpendicular torque vector and the enclosed parallelogram area.

Robotic Arm Forward and Inverse Kinematics

Industrial multi-axis articulated robotic arms translate Cartesian tool tip coordinates (x, y, z end-effector target positions) into individual rotational polar joint angles (θ₁, θ₂, θ₃) using Inverse Kinematics Trigonometric Transformations, allowing factory robots to execute smooth laser cutting and arc welding trajectories.

Complex Numbers: Euler's Formula and Polar Exponential Form

In electrical AC circuit analysis and signal processing, points in the 2D Cartesian complex plane (z = a + j·b) transform into Polar Exponential Phasor Form via Euler's Identity:

Euler's Formula: z = r × e^jθ = r × [ cos(θ) + j × sin(θ) ]
Magnitude: r = |z| = √( a² + b² ) | Phase Angle: θ = atan2( b, a )

Converting AC voltages and currents into polar phasor exponential form simplifies multiplying sinusoidal signals from complex differential calculus into simple algebraic magnitude multiplication and phase angle addition: z₁ × z₂ = ( r₁ × r₂ ) × e^{j(θ₁ + θ₂)}.

Conic Sections in Polar Equations

In orbital astrophysics and celestial mechanics, the unified geometric equation for all conic section orbits (circles, ellipses, parabolas, hyperbolas) is formulated in polar coordinates relative to a focal origin:

Polar Orbit Equation: r(θ) = [ a × ( 1 - e² ) ] / [ 1 + e × cos(θ) ]

Where e is the orbital eccentricity (e = 0 for circular orbits, 0 < e < 1 for planetary ellipses, e = 1 for parabolic escape trajectories).

Logarithmic Spirals in Nature and Engineering

In polar coordinates, the Logarithmic Spiral (r = a × e^bθ) describes self-similar geometric forms found in nautilus shells, cyclone weather radar signatures, spiral galaxy arms, and aerodynamic turbomachine impeller blades.

Cardioid and Limacon Polar Curves

In acoustics and microphone transducer design, heart-shaped directional pickup patterns are modeled by the polar equation r(θ) = a + b × cos(θ), capturing front acoustic audio while rejecting rear room reverberation.