Polar Coordinates Calculator – Guide & Formulas
Convert between polar (r, θ) and Cartesian (x, y) coordinates instantly. Get distance, angle in degrees and radians, and quadrant information.
Convert between polar coordinates (r, θ) and Cartesian coordinates (x, y) instantly. Enter either set of coordinates to get the equivalent representation, including the distance from origin, angle in both degrees and radians, and quadrant information.
Key Takeaway
Use the free Polar Coordinates Calculator to convert between polar (r, θ) and cartesian (x, y) coordinates instantly. get distance, angle in degrees and radians, and quadrant information. Get instant results with step-by-step explanations.
How to Use the Polar Coordinates Calculator
- Select the conversion direction: Cartesian to Polar or Polar to Cartesian.
- For Cartesian to Polar: enter x and y values.
- For Polar to Cartesian: enter r (distance) and θ (angle in degrees).
- Review the converted coordinates, angle in radians, and quadrant.
The Formula
Variable Definitions
- r: Radial distance from the origin (always ≥ 0)
- θ: Angular coordinate measured from the positive x-axis
- x: Horizontal Cartesian coordinate
- y: Vertical Cartesian coordinate
Convert (3, 4) from Cartesian to Polar
Find the polar coordinates of the Cartesian point (3, 4).
- Step 1: Calculate r = √(3² + 4²) = √(9 + 16) = √25 = 5.
- Step 2: Calculate θ = arctan(4/3) = arctan(1.3333) = 53.13°.
- Step 3: Verify quadrant: x = 3 > 0, y = 4 > 0, so the point is in Quadrant I.
- Step 4: Polar coordinates: (r, θ) = (5, 53.13°) or (5, 0.9273 radians).
Frequently Asked Questions
What are polar coordinates?
Polar coordinates represent a point in the plane using a distance r from the origin and an angle θ from the positive x-axis. They are written as (r, θ) and are especially useful for circular and spiral patterns.
How do I convert from Cartesian to polar?
Use r = √(x² + y²) for the distance and θ = atan2(y, x) for the angle. The atan2 function correctly handles all four quadrants, unlike a simple arctan(y/x) which only works in Quadrants I and IV.
Why use atan2 instead of arctan?
atan2(y, x) considers the signs of both x and y to determine the correct quadrant. A simple arctan(y/x) cannot distinguish between Quadrants I and III or II and IV, leading to angles off by 180°.
Can r be negative in polar coordinates?
Conventionally r ≥ 0. A negative r value means you go in the opposite direction of the angle θ (equivalent to adding 180° to θ and using positive r). This calculator uses r ≥ 0.
How do I convert polar to Cartesian?
Use x = r·cos(θ) and y = r·sin(θ). Make sure θ is in the correct unit (degrees or radians) matching your calculator settings.
What is the angle range for θ?
The angle θ is typically expressed in the range (-180°, 180°] or [0°, 360°). This calculator outputs θ in (-180°, 180°] and also provides the equivalent [0°, 360°) value.
Where are polar coordinates used?
Polar coordinates are used in navigation (bearing and distance), physics (circular motion, central forces), engineering (antenna patterns), and mathematics (spirals, roses, cardioids).
What is the relationship between polar and complex numbers?
A complex number z = x + iy can be written in polar form as z = r(cos θ + i sin θ) = r·e^(iθ), where r = |z| is the modulus and θ = arg(z) is the argument.