Polar to Rectangular Coordinates Calculator
Convert between polar (r, φ) and rectangular (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 (φ): 3.1416 radians = 180°
Common angle: 180° (π, negative x-axis)
Rectangular Coordinates
X Coordinate: -5
Y Coordinate: 0
Quadrant: II (negative x, positive y)
Trigonometric Values
Distance from Origin: 5
cos(φ): -1
sin(φ): 0
tan(φ): 0
Conversion Formulas
Polar to Rectangular:
• x = r × cos(φ)
• y = r × sin(φ)
Rectangular to Polar:
• r = √(x² + y²)
• φ = atan2(y, x)
Step-by-Step Calculation
φ = 3.1416 radians
x = 5 × cos(3.1416)
x = 5 × -1 = -5
y = 5 × sin(3.1416)
y = 5 × 0 = 0
Example Calculations
Example 1: Classic Case
Polar: (5, π) = (5, 180°)
Calculation:
x = 5 × cos(π) = 5 × (-1) = -5
y = 5 × sin(π) = 5 × 0 = 0
Rectangular: (-5, 0)
Example 2: 45° Angle
Polar: (√8, π/4) = (2.828, 45°)
Calculation:
x = √8 × cos(π/4) = √8 × (√2/2) = 2
y = √8 × sin(π/4) = √8 × (√2/2) = 2
Rectangular: (2, 2)
Example 3: Third Quadrant
Polar: (3, 7π/4) = (3, 315°)
Calculation:
x = 3 × cos(7π/4) = 3 × (√2/2) ≈ 2.121
y = 3 × sin(7π/4) = 3 × (-√2/2) ≈ -2.121
Rectangular: (2.121, -2.121)
Example 4: Y-axis Point
Polar: (4, π/2) = (4, 90°)
Calculation:
x = 4 × cos(π/2) = 4 × 0 = 0
y = 4 × sin(π/2) = 4 × 1 = 4
Rectangular: (0, 4)
Coordinate Systems
Polar Coordinates
Components: (r, φ)
r: Distance from origin
φ: Angle from positive x-axis
Perspective: Circular view
Rectangular Coordinates
Components: (x, y)
x: Horizontal position
y: Vertical position
Perspective: Grid-based view
Conversion Formulas
X Coordinate
x = r × cos(φ)
Y Coordinate
y = r × sin(φ)
Radius
r = √(x² + y²)
Angle
φ = atan2(y, x)
Calculator Tips
Radius r must be non-negative
Angles can be in radians or degrees
Common angles have exact values
Every polar point has one rectangular equivalent
Coordinates describe the same point differently
Understanding Polar and Rectangular Coordinates
The Polar Perspective
Polar coordinates see the world in terms of circles. To locate a point, imagine drawing a circle centered at the origin that passes through your point. The polar system records two pieces of information: how far the point is from the center (radius r) and what angle the line from center to point makes with the positive x-axis (angle φ).
The Rectangular Perspective
Rectangular coordinates, on the other hand, see the world as a grid. They describe a point's position by noting how far it is from the vertical axis (x-coordinate) and how far it is from the horizontal axis (y-coordinate). This creates a rectangular grid system.
Why Convert Between Systems?
Use Polar When:
- •Working with rotational motion
- •Dealing with circular patterns
- •Analyzing wave functions
- •Navigation and radar systems
Use Rectangular When:
- •Working with linear equations
- •Creating computer graphics
- •Mapping and surveying
- •Most algebraic operations
Mathematical Foundation
The conversion formulas come directly from trigonometry and the unit circle. When you have a point at distance r from the origin, making angle φ with the positive x-axis, you can project this point onto the x and y axes to get the rectangular coordinates.
X-coordinate (horizontal projection):
The x-coordinate represents how far right (positive) or left (negative) the point is from the y-axis.
x = r × cos(φ)
Y-coordinate (vertical projection):
The y-coordinate represents how far up (positive) or down (negative) the point is from the x-axis.
y = r × sin(φ)