Polar to Rectangular Converter

How to Use This Calculator

1. Select Polar to Rectangular or Rectangular to Polar, enter values, choose degree or radian — result appears instantly.
2. Use Cylindrical 3D tab to add a z-coordinate.
3. Professional shows spherical coordinates and the integration Jacobian.

Formula

Example

Frequently Asked Questions

• How do you convert polar to rectangular coordinates? ▼
  To convert from polar (r, θ) to rectangular (x, y), use x = r·cos(θ) and y = r·sin(θ). Here r is the distance from the origin (radius) and θ is the angle measured counterclockwise from the positive x-axis. Example: (r = 5, θ = 53.13°). x = 5·cos(53.13°) = 5 × 0.6 = 3. y = 5·sin(53.13°) = 5 × 0.8 = 4. So the rectangular coordinates are (3, 4). Common pitfall: using degrees when your calculator is in radian mode — always verify the angle unit. Also, r can be negative in polar form; (−3, 60°) points in the opposite direction, which is equivalent to (3, 240°).
• How do you convert rectangular to polar coordinates? ▼
  To convert from rectangular (x, y) to polar (r, θ), use r = √(x² + y²) and θ = atan2(y, x). Example: (x = 3, y = 4). r = √(9 + 16) = √25 = 5. θ = atan2(4, 3) ≈ 53.13°. So the polar form is (5, 53.13°). For a point in the second quadrant, e.g., (−3, 4): r = 5, θ = atan2(4, −3) ≈ 126.87°. A common pitfall is using arctan(y/x) directly — this only gives angles in the range (−90°, 90°) and cannot distinguish between a point in the first quadrant and its opposite in the third quadrant. Always use atan2(y, x) to get the correct quadrant.
• What is atan2 and why use it instead of atan? ▼
  atan2(y, x) is a two-argument arctangent function that returns the angle in the full range (−π, π], correctly accounting for the signs of both x and y to determine the quadrant. By contrast, atan(y/x) only returns values in (−π/2, π/2) and fails when x = 0 (division by zero) and gives wrong quadrants for negative x. Example: point (−1, −1) is in the third quadrant, angle = 225° = −135°. atan(−1/−1) = atan(1) = 45° — wrong quadrant. atan2(−1, −1) = −135° — correct. All modern programming languages and calculators provide atan2. A common pitfall is calling atan2 with the arguments in the wrong order — it is always atan2(y, x), not atan2(x, y).
• What are cylindrical coordinates? ▼
  Cylindrical coordinates (r, θ, z) extend 2D polar coordinates to three dimensions by adding a vertical z-axis. The conversion to rectangular is: x = r·cos(θ), y = r·sin(θ), z = z (unchanged). The reverse conversion is: r = √(x² + y²), θ = atan2(y, x), z = z. Example: (r = 5, θ = 53.13°, z = 7) → (x = 3, y = 4, z = 7). Cylindrical coordinates are natural for problems with rotational symmetry around the z-axis — such as electric fields of long wires, fluid flow in pipes, or magnetic fields of solenoids. A common pitfall is confusing cylindrical with spherical coordinates — in cylindrical, r is the horizontal distance from the z-axis, while in spherical, ρ is the full 3D distance from the origin.
• What is the Jacobian for polar integration? ▼
  When changing variables from rectangular (x, y) to polar (r, θ) in a double integral, the area element transforms as dA = dx·dy = r·dr·dθ. The extra factor r is the Jacobian of the transformation — it accounts for the fact that polar grid cells near the origin are smaller than those far from it. Example: to integrate a function over a disk of radius R, use ∫₀^(2π) ∫₀^R f(r,θ)·r dr dθ. Without the r factor, the integral would undercount area near the rim and overcount near the center. For spherical coordinates, the volume element is dV = ρ²·sin(φ)·dρ·dφ·dθ. Common pitfall: omitting the Jacobian when switching to polar coordinates, leading to answers off by an r-dependent factor.

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