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Polar to Cartesian Conversion

Convert polar (r, θ) to Cartesian (x, y) using x = r cos θ and y = r sin θ. θ must be in radians for cos/sin. Used in physics, graphics, and complex numbers.

Concept Fundamentals
x = r cos θ
x
y = r sin θ
y
r=1 → (cos θ, sin θ)
Unit
θ in rad for cos/sin
Radians

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Unit circle: r=1 gives (cos θ, sin θ). Euler: e^(iθ) = cos θ + i sin θ. cos and sin handle all quadrants automatically.

Key quantities
x = r cos θ
x
Key relation
y = r sin θ
y
Key relation
r=1 → (cos θ, sin θ)
Unit
Key relation
θ in rad for cos/sin
Radians
Key relation

Ready to run the numbers?

Why: Polar-to-Cartesian conversion is essential for circular motion, orbital paths, game sprites, and complex numbers (reiθ = r(cos θ + i sin θ)).

How: Apply x = r cos θ and y = r sin θ. If θ is in degrees, convert first: θ_rad = θ_deg × π/180. When r = 0, (x,y) = (0,0) regardless of θ.

Unit circle: r=1 gives (cos θ, sin θ).Euler: e^(iθ) = cos θ + i sin θ.
Sources:Khan Academy — Polar CoordinatesWolfram MathWorld — Polar Coordinates

Run the calculator when you are ready.

Convert Polar to CartesianEnter r and θ

Enter Polar Coordinates

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🧮 Fascinating Math Facts

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x = r cos θ, y = r sin θ.

— Conversion

○

r=1 traces the unit circle.

— Trigonometry

Key Takeaways

  • • x = r · cos(θ) and y = r · sin(θ)
  • • θ must be in radians when using cos/sin in most programming and math contexts
  • • For θ in degrees, convert first: θ_rad = θ_deg × π/180
  • • When r = 0, (x, y) = (0, 0) regardless of θ
  • • The unit circle: r = 1 gives points (cos θ, sin θ) on the circle of radius 1

Did You Know?

Unit Circle

When r = 1, polar coordinates trace the unit circle. (cos θ, sin θ) is the standard parametrization.

Euler's Formula

e^(iθ) = cos θ + i sin θ. Polar (r,θ) corresponds to the complex number r·e^(iθ).

Trigonometry Origin

The definitions cos θ = x/r and sin θ = y/r come directly from the right triangle formed by (x,y) and the origin.

Periodicity

cos and sin are 2π-periodic. (r, θ) and (r, θ + 2π) give the same Cartesian point.

Negative r

If r < 0, (r,θ) = (|r|, θ+π) in many conventions. This calculator uses r ≥ 0.

Graphics & Games

Polar-to-Cartesian is used for circular motion, orbital paths, and rotating sprites in game engines.

Understanding Polar to Cartesian Conversion

From polar (r, θ), drop a perpendicular to the x-axis. The horizontal leg is x = r·cos(θ), the vertical leg is y = r·sin(θ).

x=rcostheta,y=rsinthetax = r \\cos\\theta, \quad y = r \\sin\\theta

Ensure θ is in radians when evaluating cos and sin. Degrees must be converted: θ_rad = θ_deg × π/180.

Expert Tips

Radians vs Degrees

Most math libraries (JavaScript, Python, etc.) expect angles in radians. Always convert degrees to radians before calling cos/sin.

Check Quadrants

cos and sin automatically handle all quadrants. cos(θ) is positive in I and IV, sin(θ) in I and II.

Unit Circle Values

Memorize: (1,0°), (√2/2,45°), (0,90°), (-√2/2,135°), (-1,180°). These help verify calculations.

r = 0

When r = 0, you get the origin (0,0) regardless of θ. The angle is irrelevant at the origin.

Frequently Asked Questions

What are the formulas for x and y?

x = r·cos(θ) and y = r·sin(θ). The angle θ must be in radians for these formulas.

Can I use degrees instead of radians?

Yes, but convert first: θ_radians = θ_degrees × π/180. This calculator supports both modes.

What if r is negative?

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

What is the unit circle?

When r = 1, (cos θ, sin θ) traces a circle of radius 1 centered at the origin.

How do I convert back to polar?

Use r = √(x²+y²) and θ = atan2(y, x).

Why does (1, 90°) give (0, 1)?

At 90°, the point is on the positive y-axis. cos(90°)=0, sin(90°)=1, so (0, 1).

What about θ = 180°?

cos(180°)=-1, sin(180°)=0, so (r, 180°) = (-r, 0) — the point is on the negative x-axis.

How to Use This Calculator

  1. Enter r (radius) and θ (angle). Select degrees or radians for the angle.
  2. Click a sample example to auto-fill and calculate, or enter your own values.
  3. Click Calculate to get Cartesian (x, y).
  4. Review the visualization and step-by-step solution.
  5. Use Copy Results to share.

Note: Angles in radians or degrees are supported. Results use standard floating-point arithmetic.

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