Polar Coordinates Calculator
Convert between Cartesian (x, y) and polar (r, θ) coordinate systems with step-by-step solutions
Coordinate Conversion Calculator
Cartesian Coordinates (x, y)
Conversion Results
Input: Cartesian Coordinates
(x, y) = (3, 4)
Location: Quadrant I
Result: Polar Coordinates
(r, θ) = (5.0000, 53.13°)
Radius: r = 5.0000
Angle (degrees): θ = 53.13°
Angle (radians): θ = 0.9273 rad
Coordinate Visualization
Step-by-Step Solution
1. Calculate the radius (r)
Formula: r = √(x² + y²)
Substitute: r = √((3)² + (4)²)
Calculate: r = √(9 + 16)
Result: r = √25 = 5.0000
2. Calculate the angle (θ)
Formula: θ = arctan(y/x) using atan2 for correct quadrant
Substitute: θ = atan2(4, 3)
Result (radians): θ = 0.9273 rad
Result (degrees): θ = 53.13°
3. Verify the quadrant
Point (3, 4) is located in: Quadrant I
The angle θ = 53.13° is in the correct quadrant
4. Final polar coordinates
(r, θ) = (5.0000, 53.13°)
Example Conversions
Cartesian to Polar
Input: (3, 4)
Radius: r = √(3² + 4²) = 5
Angle: θ = arctan(4/3) = 53.13°
Result: (5, 53.13°)
Polar to Cartesian
Input: (2, 90°)
x: x = 2 × cos(90°) = 0
y: y = 2 × sin(90°) = 2
Result: (0, 2)
Conversion Formulas
Cartesian → Polar
r = √(x² + y²)
θ = arctan(y/x)
Polar → Cartesian
x = r × cos(θ)
y = r × sin(θ)
Angle Conversion
Rad = Deg × π/180
Deg = Rad × 180/π
Calculator Tips
Radius (r) is always non-negative
Angle θ is typically in range (-π, π] or (-180°, 180°]
Use atan2 function for correct quadrant determination
Origin (0,0) has r=0 and undefined angle
Understanding Coordinate Systems
Cartesian Coordinates
The Cartesian coordinate system uses two perpendicular axes (x and y) to locate points in a plane. Any point is described by its distances from these axes.
Key Features
- •Origin: Point (0, 0) where axes intersect
- •X-axis: Horizontal axis (left-right)
- •Y-axis: Vertical axis (up-down)
- •Quadrants: Four regions divided by axes
Polar Coordinates
The polar coordinate system uses a distance from the origin (radius r) and an angle (θ) from a reference direction to locate points.
Key Features
- •Pole: Central point (origin)
- •Polar axis: Reference direction (usually positive x-axis)
- •Radius (r): Distance from pole
- •Angle (θ): Direction from polar axis
When to Use Each System
Cartesian System
- • Linear relationships
- • Rectangular grids
- • Computer graphics
- • Engineering drawings
Polar System
- • Circular motion
- • Rotational mechanics
- • Radar systems
- • Complex numbers
Real-World Applications
Navigation
GPS systems, aircraft positioning, maritime navigation using coordinate conversions
Robotics
Robot arm control, path planning, sensor data processing in different coordinate systems
Physics
Circular motion analysis, wave functions, electromagnetic field calculations