Real Rectangular Polar Calculator

This calculator converts between rectangular (Cartesian) coordinates (x, y) and polar coordinates (r, θ). It's useful for physics, engineering, and computer graphics applications where coordinate systems need to be transformed.

What is Rectangular to Polar Conversion?

Rectangular coordinates (x, y) describe a point's position using horizontal and vertical distances from a reference point. Polar coordinates (r, θ) describe a point using a distance from the origin (r) and an angle (θ) from a reference direction.

In mathematics, the rectangular coordinate system is also called Cartesian coordinates, while polar coordinates are sometimes called polar or cylindrical coordinates.

The conversion between these systems is fundamental in many scientific and engineering fields. For example, in robotics, polar coordinates help describe circular paths, while rectangular coordinates are more intuitive for linear movements.

How to Use This Calculator

Select whether you're converting from rectangular to polar or vice versa
Enter the known coordinate values
Click "Calculate" to see the converted coordinates
Review the result and chart visualization
Use the "Reset" button to start a new calculation

To convert rectangular to polar coordinates:

r = √(x² + y²)

θ = arctan(y/x)

Conversion Formulas

The exact formulas used in this calculator are:

Rectangular to Polar Conversion

r = √(x² + y²)

θ = arctan(y/x) (in radians)

θ = arctan(y/x) × (180/π) (in degrees)

Polar to Rectangular Conversion

x = r × cos(θ)

y = r × sin(θ)

Note that the angle θ is measured from the positive x-axis in the standard mathematical convention.

Worked Examples

Example 1: Rectangular to Polar

Convert rectangular coordinates (3, 4) to polar coordinates.

Calculate r: √(3² + 4²) = √(9 + 16) = √25 = 5
Calculate θ: arctan(4/3) ≈ 0.927 radians (53.13°)

Result: Polar coordinates (5, 0.927 radians)

Example 2: Polar to Rectangular

Convert polar coordinates (5, 0.927 radians) to rectangular coordinates.

Calculate x: 5 × cos(0.927) ≈ 3
Calculate y: 5 × sin(0.927) ≈ 4

Result: Rectangular coordinates (3, 4)

Coordinate Conversion Comparison
Rectangular (x, y)	Polar (r, θ in radians)
(1, 1)	(1.414, 0.785)
(0, 5)	(5, 1.571)
(-3, 4)	(5, 2.214)

FAQ

What is the difference between rectangular and polar coordinates?: Rectangular coordinates describe a point's position using horizontal and vertical distances from a reference point, while polar coordinates describe a point using a distance from the origin and an angle from a reference direction.
When would I use rectangular vs. polar coordinates?: Use rectangular coordinates for linear measurements and Cartesian geometry. Use polar coordinates for circular paths, rotational motion, and radial measurements.
How do I handle negative coordinates in the conversion?: The calculator automatically handles negative values by adjusting the angle θ to the correct quadrant. For example, (-3, 4) converts to (5, 2.214 radians) in the second quadrant.
What units should I use for the angle θ?: The calculator accepts angles in both radians and degrees. Make sure to select the correct unit in the input field.
Is this calculator accurate for all real numbers?: Yes, the calculator uses standard mathematical functions that work with all real numbers. However, very large numbers may cause precision issues due to floating-point arithmetic limitations.