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Cartesian to Polar

Convert (x,y) to (r,θ): r = √(x²+y²) is distance from origin; θ = atan2(y,x) is angle from positive x-axis. Use atan2 for correct quadrant. Essential for circular motion, waves, and complex numbers.

Concept Fundamentals
√(x²+y²)
r
atan2(y,x)
θ
x=r cos θ, y=r sin θ
Inverse
r=0, θ undefined
Origin

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atan2(y,x) returns angles in (-π, π]; add 2π for [0, 2π). Inverse conversion: x = r·cos(θ), y = r·sin(θ). Some conventions allow negative r: (-r,θ) = (r,θ+π).

Key quantities
√(x²+y²)
r
Key relation
atan2(y,x)
θ
Key relation
x=r cos θ, y=r sin θ
Inverse
Key relation
r=0, θ undefined
Origin
Key relation

Ready to run the numbers?

Why: Polar coordinates simplify circular motion, waves, antenna patterns, radar, and complex numbers. Euler's formula: r·e^(iθ) = r(cos θ + i sin θ).

How: r = √(x²+y²), θ = atan2(y,x). Never use atan(y/x) alone—it fails in quadrants II and III. At origin (0,0), r=0 and θ is undefined.

atan2(y,x) returns angles in (-π, π]; add 2π for [0, 2π).Inverse conversion: x = r·cos(θ), y = r·sin(θ).
Sources:Khan Academy — Polar CoordinatesWolfram MathWorld — Polar Coordinates

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Convert CoordinatesEnter (x, y) to get (r, θ)

Enter Cartesian Coordinates

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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r = √(x²+y²), θ = atan2(y,x).

— Coordinate Geometry

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Inverse: x = r cos θ, y = r sin θ.

— Conversion

Key Takeaways

  • • r is the radial distance: r=x2+y2r = \sqrt{x^2 + y^2}
  • • θ is the angle from the positive x-axis: θ=atan2(y,x)\theta = \text{atan2}(y, x)
  • • Use atan2(y, x) instead of atan(y/x) to get the correct quadrant
  • • At the origin (0,0), r = 0 and θ is undefined
  • • Polar coordinates are ideal for circular motion, waves, and rotational symmetry

Did You Know?

Physics & Circular Motion

Polar coordinates simplify equations for orbits, pendulums, and rotating systems. Newton's laws in polar form describe planetary motion elegantly.

Complex Numbers

Euler's formula connects polar and Cartesian: a point (r,θ) corresponds to the complex number r·e^(iθ) = r(cos θ + i sin θ).

Navigation

Radar and sonar systems use polar coordinates: distance (r) and bearing (θ) describe target positions.

Computer Graphics

Sprites and particles often use polar coordinates for circular paths, spirals, and radial gradients.

Engineering

Stress analysis in circular structures (pipes, shafts) uses cylindrical coordinates, which extend polar to 3D.

Quadrant Ambiguity

atan(y/x) alone gives wrong angles in quadrants II and III. atan2(y,x) handles all four quadrants correctly.

Understanding Cartesian to Polar Conversion

Cartesian (x, y) uses perpendicular distances. Polar (r, θ) uses distance from origin and angle. The conversion formulas:

r=x2+y2,θ=atan2(y,x)r = \sqrt{x^2 + y^2}, \quad \theta = \text{atan2}(y, x)

atan2 returns angles in (-π, π]; add 2π if you need [0, 2π).

Expert Tips

Use atan2

Never use atan(y/x) alone — it fails in quadrants II and III. atan2(y,x) gives the correct angle in all quadrants.

Origin Special Case

At (0,0), r=0 but θ is undefined. Many conventions use θ=0 by default, but it's arbitrary.

Angle Conventions

Angles are usually measured counterclockwise from the positive x-axis. Check your textbook or software for conventions.

Negative r

Some conventions allow negative r: (r,θ) = (-r, θ+π). This calculator uses r ≥ 0.

Frequently Asked Questions

What is the formula for r?

r = √(x² + y²). It is the distance from the origin to the point (x, y), using the Pythagorean theorem.

Why use atan2 instead of atan?

atan(y/x) only returns angles in (-π/2, π/2). atan2(y,x) returns the full range (-π, π] and correctly identifies the quadrant.

What happens at the origin?

At (0,0), r = 0. The angle θ is undefined because there is no unique direction from the origin.

How do I convert θ from radians to degrees?

Multiply by 180/π. For example, π/4 radians = 45°.

What are polar coordinates used for?

Circular motion, waves, antenna patterns, radar, complex numbers, and any problem with rotational symmetry.

Can r be negative?

In some conventions, yes: (-r, θ) = (r, θ+π). This calculator uses r ≥ 0.

How do I convert back to Cartesian?

Use x = r·cos(θ) and y = r·sin(θ). Ensure θ is in radians for these formulas.

How to Use This Calculator

  1. Enter the Cartesian coordinates (x, y), or click a sample example to auto-fill and calculate.
  2. Click Calculate to get polar coordinates (r, θ) in both radians and degrees.
  3. Review the visualization showing the point, radius, and angle on the coordinate plane.
  4. Examine the step-by-step solution for the derivation.
  5. Use Copy Results to share or paste into assignments.

Note: This calculator uses standard floating-point arithmetic. For educational and professional use. Angle θ is normalized to [0°, 360°).

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