Polar to Cartesian Coordinates Calculator
Convert between polar (r, θ) and Cartesian (x, y) coordinate systems
Coordinate Conversion
Distance from origin (must be ≥ 0)
Angle from positive x-axis
Conversion Results
Polar Coordinates
Radius (r): 5
Angle (θ): 0.7854 radians = 45°
Cartesian Coordinates
X Coordinate: 3.5355
Y Coordinate: 3.5355
Quadrant: I (positive x, positive y)
Additional Information
Distance from Origin: 5
cos(θ): 0.7071
sin(θ): 0.7071
tan(θ): 1
Conversion Formulas
Polar to Cartesian:
• x = r × cos(θ)
• y = r × sin(θ)
Cartesian to Polar:
• r = √(x² + y²)
• θ = atan2(y, x)
Step-by-Step Calculation
θ = 0.7854 radians
x = r × cos(θ) = 5 × cos(0.7854)
x = 5 × 0.7071 = 3.5355
y = r × sin(θ) = 5 × sin(0.7854)
y = 5 × 0.7071 = 3.5355
Example Calculations
Example 1: Unit Circle
Polar: (1, π/4) = (1, 45°)
Calculation:
x = 1 × cos(π/4) = 1 × √2/2 ≈ 0.707
y = 1 × sin(π/4) = 1 × √2/2 ≈ 0.707
Cartesian: (0.707, 0.707)
Example 2: Second Quadrant
Polar: (2, 3π/4) = (2, 135°)
Calculation:
x = 2 × cos(3π/4) = 2 × (-√2/2) ≈ -1.414
y = 2 × sin(3π/4) = 2 × (√2/2) ≈ 1.414
Cartesian: (-1.414, 1.414)
Example 3: Negative Coordinates
Cartesian: (-3, -4)
Calculation:
r = √((-3)² + (-4)²) = √(9 + 16) = 5
θ = atan2(-4, -3) ≈ -2.214 rad ≈ 233.13°
Polar: (5, 233.13°)
Example 4: On Axis
Polar: (5, π/2) = (5, 90°)
Calculation:
x = 5 × cos(π/2) = 5 × 0 = 0
y = 5 × sin(π/2) = 5 × 1 = 5
Cartesian: (0, 5)
Coordinate Systems
Polar Coordinates
Components: (r, θ)
r: Distance from origin
θ: Angle from positive x-axis
Best for: Circular motion, rotations
Cartesian Coordinates
Components: (x, y)
x: Horizontal position
y: Vertical position
Best for: Linear motion, grids
Conversion Formulas
X Coordinate
x = r × cos(θ)
Y Coordinate
y = r × sin(θ)
Radius
r = √(x² + y²)
Angle
θ = atan2(y, x)
Calculator Tips
Radius r is always non-negative
Angle θ can be in radians or degrees
atan2 function handles all quadrants
Origin is (0,0) in both systems
Angles are measured counterclockwise
Understanding Coordinate Systems
What are Polar Coordinates?
Polar coordinates describe a point's position using its distance from the origin (radius r) and the angle θ measured counterclockwise from the positive x-axis. This system is particularly useful for describing circular motion, spirals, and rotational phenomena.
What are Cartesian Coordinates?
Cartesian coordinates use two perpendicular axes (x and y) to specify a point's location. The x-coordinate represents horizontal displacement, while the y-coordinate represents vertical displacement from the origin.
When to Use Each System?
Use Polar Coordinates for:
- •Circular and rotational motion
- •Periodic functions and oscillations
- •Curves with radial symmetry
- •Navigation and radar systems
Use Cartesian Coordinates for:
- •Linear motion and straight lines
- •Rectangular grids and maps
- •Computer graphics and screen coordinates
- •Mathematical functions and graphs
Understanding the atan2 Function
The atan2(y, x) function is crucial for converting from Cartesian to polar coordinates. Unlike the regular arctan function, atan2 considers the signs of both x and y to determine the correct quadrant and returns angles in the range [-π, π].
Why not just use arctan(y/x)?
The arctan function only returns values in [-π/2, π/2], which means it can't distinguish between:
- • Points in quadrants I and III
- • Points in quadrants II and IV
- • Cases where x = 0 (division by zero)
atan2 handles all cases:
- • Quadrant I: atan2(+y, +x) → [0, π/2]
- • Quadrant II: atan2(+y, -x) → [π/2, π]
- • Quadrant III: atan2(-y, -x) → [-π, -π/2]
- • Quadrant IV: atan2(-y, +x) → [-π/2, 0]
- • Special cases: atan2(0, 0) = 0