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Cylindrical to Cartesian

Convert (r,θ,z) to (x,y,z): x = r cos θ, y = r sin θ, z = z. Same as polar-to-Cartesian in the xy-plane with z unchanged. Use radians for cos/sin.

Concept Fundamentals
r cos θ
x
r sin θ
y
z unchanged
z
r=√(x²+y²), θ=atan2(y,x)
Inverse

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Identical to 2D polar: (r,θ) → (r cos θ, r sin θ). r ≥ 0. Negative r can be handled as (|r|, θ+π). Inverse: r = √(x²+y²), θ = atan2(y,x), z = z.

Key quantities
r cos θ
x
Key relation
r sin θ
y
Key relation
z unchanged
z
Key relation
r=√(x²+y²), θ=atan2(y,x)
Inverse
Key relation

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Why: Cylindrical-to-Cartesian conversion is needed when combining cylindrical solutions (e.g., from Laplace equation) with Cartesian boundary conditions, or for visualization.

How: x = r cos θ, y = r sin θ, z = z. Ensure θ is in radians when using cos/sin. For degrees, convert first: θ_rad = θ_deg × π/180.

Identical to 2D polar: (r,θ) → (r cos θ, r sin θ).r ≥ 0. Negative r can be handled as (|r|, θ+π).
Sources:Khan Academy — Cylindrical CoordinatesWolfram MathWorld — Cylindrical Coordinates

Run the calculator when you are ready.

Convert to CartesianEnter (r, θ, z) to get (x, y, z)

Enter Cylindrical Coordinates

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

🧮 Fascinating Math Facts

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x = r cos θ, y = r sin θ, z = z.

— 3D Geometry

θ

Use radians for cos/sin in these formulas.

— Note

Key Takeaways

  • • x = r · cos(θ), y = r · sin(θ), z = z
  • • Same as polar-to-Cartesian in the xy-plane; z is unchanged
  • • θ must be in radians when evaluating cos and sin
  • • When r = 0, (x, y) = (0, 0) regardless of θ
  • • Cylindrical is ideal for problems with axial symmetry

Did You Know?

Polar Extension

Cylindrical is polar (r,θ) in the xy-plane plus z. The x,y formulas are identical to 2D polar-to-Cartesian.

Maxwell Equations

Electromagnetic problems with cylindrical symmetry (coaxial cables, solenoids) use cylindrical coordinates.

Heat Transfer

Radial heat flow in pipes and cylindrical shells is modeled in cylindrical coordinates.

r = 0

When r = 0, the point is on the z-axis. x = y = 0 regardless of θ.

Inverse Formulas

To convert Cartesian to cylindrical: r = √(x²+y²), θ = atan2(y,x), z = z.

Unit Vectors

Cylindrical unit vectors êᵣ, êθ, êz relate to Cartesian via êᵣ = cos θ î + sin θ ĵ.

Understanding Cylindrical to Cartesian

Project (r,θ) onto the xy-plane: x = r·cos(θ), y = r·sin(θ). The z-coordinate is unchanged.

x=rcostheta,y=rsintheta,z=zx = r \\cos\\theta, \quad y = r \\sin\\theta, \quad z = z

Ensure θ is in radians when evaluating cos and sin.

Expert Tips

Radians Required

cos and sin expect radians. Convert degrees: θ_rad = θ_deg × π/180.

r = 0

When r = 0, you get (0, 0, z) — a point on the z-axis. θ is irrelevant.

Verify with r

Check: x² + y² = r². Use this to verify your conversion.

Angle Periodicity

(r, θ) and (r, θ + 2π) give the same (x, y). Angles are 2π-periodic.

Frequently Asked Questions

What are the conversion formulas?

x = r·cos(θ), y = r·sin(θ), z = z. θ must be in radians for cos and sin.

Can I use degrees for θ?

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

What happens when r = 0?

You get (0, 0, z) — a point on the z-axis. The angle θ does not affect the result.

How do I convert Cartesian to cylindrical?

Use r = √(x²+y²), θ = atan2(y,x), z = z. See our Cartesian to Cylindrical calculator.

Why does z stay the same?

Both coordinate systems use the same z-axis. z measures height above the xy-plane in both.

What is cylindrical used for?

Problems with cylindrical symmetry: wires, pipes, rotating fluids, pressure vessels.

How do I verify my result?

Check that x² + y² = r² and that (x,y) lies at angle θ from the x-axis.

How to Use This Calculator

  1. Enter cylindrical (r, θ, z). Select degrees or radians for θ.
  2. Click a sample example to auto-fill and calculate.
  3. Click Calculate to get Cartesian (x, y, z).
  4. Review the step-by-step solution and use Copy Results to share.

Note: Standard floating-point arithmetic. r must be non-negative.

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