Cartesian to Polar Converter
Convert Cartesian to Polar
Convert Cartesian coordinates (x,y) to Polar coordinates (r,θ). Calculate radius, angle in degrees/radians, and visualize on coordinate plane.
Polar Coordinates Result
(5.00, 53.13°)
Coordinate Visualization
Cartesian Quadrant
Point is in Quadrant I (x>0, y>0)
Conversion Formulas:
θ = atan2(y, x)
atan2(y,x) gives correct quadrant for θ
Distance from Origin
Euclidean distance = √(x² + y²)
Also called magnitude or modulus
Step-by-Step Calculation:
Angle Conversions
Complex Number Form
Polar form: 5.00 e^(i·0.927)
Euler's formula: r·e^(iθ) = r(cosθ + i·sinθ)
Equivalent Coordinates: (5, 413.13°), (5, -306.87°)
Slope: 1.333 (4/3)
Applications: Physics, engineering, computer graphics, navigation
Cartesian coordinates (3, 4) convert to polar coordinates (5, 53.13°). The point is 5 units from the origin at an angle of 53.13° from the positive x-axis, located in Quadrant I.
What are Cartesian and Polar Coordinates?
Cartesian coordinates (x, y) represent points in a plane using perpendicular x and y axes. Polar coordinates (r, θ) represent points using distance from origin (radius r) and angle from positive x-axis (θ). Cartesian is ideal for rectangular systems, while polar excels for circular and rotational systems.
Coordinate System Comparison
Cartesian System
Rectangular grid
Perpendicular axes
Polar System
Circular grid
Radius and angle
Cylindrical
3D extension
Adds height z
Spherical
3D radial
Two angles, radius
Conversion Formulas
1. Cartesian to Polar
θ = atan2(y, x)
Where:
- r = radius (distance from origin)
- θ = angle in radians
- x, y = Cartesian coordinates
- atan2(y,x) = arctangent function with quadrant correction
2. Special Cases and Quadrants
Quadrant II (x<0, y>0): θ = arctan(y/x) + π
Quadrant III (x<0, y<0): θ = arctan(y/x) + π
Quadrant IV (x>0, y<0): θ = arctan(y/x) + 2π
3. Polar to Cartesian (Inverse)
y = r·sin(θ)
Where:
- x, y = Cartesian coordinates
- r = polar radius
- θ = polar angle in radians
- cos, sin = trigonometric functions
Common Coordinate Conversions
| Cartesian (x, y) | Polar (r, θ) | Quadrant | Special Property |
|---|---|---|---|
| (3, 4) | (5, 53.13°) | I | 3-4-5 right triangle |
| (-3, 4) | (5, 126.87°) | II | Mirror of (3,4) |
| (-3, -4) | (5, -126.87°) | III | Negative both axes |
| (3, -4) | (5, -53.13°) | IV | Below x-axis |
Special Points and Their Polar Forms
| Point Name | Cartesian | Polar | Significance |
|---|---|---|---|
| Origin | (0, 0) | (0, any θ) | Zero radius, angle undefined |
| Unit Circle Points | (cosθ, sinθ) | (1, θ) | Always r=1 |
| Positive X-axis | (r, 0) | (r, 0°) | Angle = 0° |
| Positive Y-axis | (0, r) | (r, 90°) | Angle = 90° |
Real-World Applications
Physics & Engineering
- Circular motion: Describing rotational systems and orbital mechanics
- Electromagnetism: Calculating electric fields around point charges
- Fluid dynamics: Analyzing flow in circular pipes and channels
- Signal processing: Representing signals in polar form (amplitude and phase)
Computer Graphics & Robotics
- 2D graphics: Rotating objects around points
- Robot navigation: Path planning in circular coordinates
- Game development: Character movement in circular arenas
- Computer vision: Radial distortion correction
Navigation & Geography
- Radar systems: Tracking objects by range and bearing
- GPS coordinates: Converting between map projections
- Marine navigation: Course plotting using bearings
- Astronomy: Celestial coordinate systems
Mathematics & Education
- Complex numbers: Euler's formula and polar representation
- Calculus: Solving integrals in circular domains
- Differential equations: Problems with radial symmetry
- Geometry: Studying curves like spirals and circles
Step-by-Step Conversion Examples
Example 1: (3, 4) to Polar
- Calculate radius: r = √(3² + 4²) = √(9 + 16) = √25 = 5
- Calculate angle: θ = atan2(4, 3)
- atan2(4,3) = arctan(4/3) = arctan(1.3333)
- arctan(1.3333) = 0.9273 radians
- Convert to degrees: 0.9273 × (180/π) = 53.13°
- Since x>0 and y>0, point is in Quadrant I
- Final polar coordinates: (5, 53.13°)
Example 2: (-5, 5) to Polar
- Calculate radius: r = √((-5)² + 5²) = √(25 + 25) = √50 = 7.07
- Calculate angle: θ = atan2(5, -5)
- atan2(5,-5) = arctan(5/-5) = arctan(-1)
- arctan(-1) = -45° or 135° (since x<0, use 135°)
- Point is in Quadrant II (x<0, y>0)
- Final polar coordinates: (7.07, 135°)
- Alternative: (7.07, 2.356 radians)
Related Calculators
Frequently Asked Questions (FAQs)
Q: What's the difference between atan and atan2?
A: atan(y/x) only gives angles between -90° and 90°, losing quadrant information. atan2(y,x) considers both x and y signs, giving correct angles in all four quadrants (-180° to 180°). Always use atan2 for coordinate conversions.
Q: How do I handle negative radius in polar coordinates?
A: In standard polar coordinates, r ≥ 0. Negative r can be converted to positive by adding 180° to θ: (-r, θ) = (r, θ+180°). Some systems allow negative r, but standard mathematics uses r ≥ 0.
Q: What happens at the origin (0,0)?
A: At the origin, r = 0 and θ is undefined (any angle works). In polar coordinates, (0, θ) represents the origin for any θ. This is the only point where angle is arbitrary.
Q: How do I convert between different angle ranges?
A: Common ranges: [-180°, 180°] (atan2 standard), [0°, 360°) (add 360° to negative angles), [0, 2π) radians. To convert: if θ < 0, add 360° (or 2π radians) to get positive angle.
Master coordinate conversions with Toolivaa's free Cartesian to Polar Converter, and explore more mathematical tools in our Coordinate Calculators collection.