Rectangular to Polar Coordinates Calculator
Convert Cartesian (x, y) coordinates to polar (r, θ) coordinates with step-by-step solutions
Convert Rectangular to Polar Coordinates
Horizontal distance from origin
Vertical distance from origin
Polar Coordinates Results
Rectangular coordinates: (0, 0)
Polar coordinates: (0.0000, 0.00°)
Quadrant: Origin
Coordinate Analysis
Enter coordinates to see detailed analysis
Example Calculation
Convert (3, 4) to Polar Coordinates
Given: Rectangular coordinates (3, 4)
Step 1: Calculate radius
r = √(3² + 4²) = √(9 + 16) = √25 = 5
Step 2: Calculate angle
θ = arctan(4/3) = arctan(1.333) = 53.13°
Result: Polar coordinates = (5, 53.13°)
Special Cases
• Origin (0, 0) → (0, 0°)
• Positive x-axis (a, 0) → (a, 0°)
• Positive y-axis (0, a) → (a, 90°)
• Negative x-axis (-a, 0) → (a, 180°)
• Negative y-axis (0, -a) → (a, 270°)
Coordinate Systems
Rectangular (Cartesian)
Point: (x, y)
Horizontal and vertical distances
Polar
Point: (r, θ)
Distance and angle from origin
Conversion Formulas
Rectangular → Polar
r = √(x² + y²)
θ = arctan(y/x)
Polar → Rectangular
x = r × cos(θ)
y = r × sin(θ)
Quick Tips
Radius is always non-negative
Angle can be measured in degrees or radians
Use atan2 function for proper quadrant determination
Multiple polar representations exist for same point
Understanding Rectangular to Polar Coordinates Conversion
What are Coordinate Systems?
Coordinate systems are mathematical tools used to specify the location of points in space. The rectangular (Cartesian) system uses perpendicular axes, while the polar system uses distance and angle measurements from a central point.
When to Use Polar Coordinates?
- •Circular or rotational motion problems
- •Oscillations and wave functions
- •Complex number representations
- •Navigation and satellite positioning
Conversion Process
Step 1: Calculate Radius
r = √(x² + y²)
Uses the Pythagorean theorem to find distance from origin
Step 2: Calculate Angle
θ = arctan(y/x) or atan2(y, x)
Uses inverse tangent with quadrant consideration
Note: The atan2 function properly handles all quadrants and special cases like division by zero when x = 0.