Polar to Cartesian Converter

Convert between polar coordinates (r, θ), Cartesian coordinates (x, y), and complex numbers. Perfect for mathematics, physics, engineering, and signal processing.

Convert between polar coordinates (r, θ) and Cartesian coordinates (x, y), including complex numbers

Value

From Unit

To Unit

Quick Examples

Cartesian to Polar

Enter 3 in "X Coordinate" and convert to "Radius (Magnitude)"

Polar to Cartesian

Enter 1 in "Radius (Magnitude)" and convert to "X Coordinate"

Common Use Cases

•Mathematics: Convert between coordinate systems for calculus and geometry
•Physics: Convert force vectors and motion analysis between coordinate systems
•Engineering: Convert design coordinates and complex number calculations
•Signal Processing: Convert between time and frequency domain representations
•Computer Graphics: Convert between different coordinate systems for rendering

Popular Conversions

(3, 4)→r = 5, θ = 0.927

(1, π/4)→x = 0.707, y = 0.707

1 + i→|z| = 1.414, arg = 0.785

(2, π/3)→x = 1, y = 1.732

How Our Polar-Cartesian Converter Works

1️⃣

Enter Coordinates

Input your coordinate values in the "From" field. Our converter accepts decimal numbers, fractions, and mathematical constants like π.

2️⃣

Select Conversion

Choose your source coordinate system and target coordinate system from our comprehensive selection including Cartesian, polar, and complex numbers.

3️⃣

Get Instant Result

View the converted coordinates in real-time. Copy results, explore common conversions, or check conversion history.

Understanding Coordinate Systems

Cartesian Coordinates (Rectangular)

X Coordinate

Horizontal distance from the y-axis. Positive values extend right, negative values extend left.

Y Coordinate

Vertical distance from the x-axis. Positive values extend upward, negative values extend downward.

Polar Coordinates

Radius (r)

Distance from the origin to the point. Always a positive value representing the magnitude.

Angle (θ)

Angle from the positive x-axis, measured counterclockwise. Typically measured in radians.

Complex Numbers

Real Part (a)

The real component of the complex number z = a + bi. Represents the x-coordinate in the complex plane.

Imaginary Part (b)

The imaginary component of the complex number z = a + bi. Represents the y-coordinate in the complex plane.

Conversion Formulas

Cartesian to Polar

r = √(x² + y²), θ = arctan(y/x). The angle θ is determined by the quadrant of the point.

Polar to Cartesian

x = r·cos(θ), y = r·sin(θ). These formulas directly convert polar coordinates to rectangular form.

Frequently Asked Questions

What is the difference between polar and Cartesian coordinates?

Cartesian coordinates use perpendicular x and y axes to locate points, while polar coordinates use a radius (distance from origin) and an angle (from the positive x-axis) to locate points. Both systems can represent the same points but in different ways.

How do I handle negative angles in polar coordinates?

Negative angles are measured clockwise from the positive x-axis. For example, -π/4 radians is equivalent to 7π/4 radians (both represent the same direction). Our converter automatically handles angle normalization.

Can I convert 3D coordinates?

Yes! Our converter supports 3D coordinates including Cartesian (x, y, z) and spherical coordinates (r, θ, φ). The z-coordinate represents depth in 3D space, and φ is the azimuthal angle from the positive z-axis.

How accurate are the complex number conversions?

Our converter uses precise mathematical formulas and maintains accuracy up to 8 decimal places. Complex number conversions include both magnitude and phase angle calculations for engineering and scientific applications.

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