Mathematics · Geometry · Analytic Geometry
Polar to Cartesian Converter
Converts polar coordinates (r, θ) to Cartesian coordinates (x, y) using standard trigonometric identities.
Calculator
Formula
r is the radial distance from the origin (must be ≥ 0 for standard convention); θ is the angle measured counterclockwise from the positive x-axis, in degrees or radians; x is the horizontal Cartesian component; y is the vertical Cartesian component.
Source: Stewart, J. Calculus: Early Transcendentals, 8th ed. Cengage Learning. Section 10.3: Polar Coordinates.
How it works
Polar coordinates describe a point in the plane by two quantities: the radial distance r from the origin and the angle θ measured counterclockwise from the positive x-axis. While polar coordinates are natural for circular motion, spirals, and wave phenomena, most algebraic and computational tools operate in Cartesian (x, y) form. Converting between the two systems is therefore an essential skill across mathematics and its applications.
The conversion formulas derive directly from the definition of sine and cosine in a right triangle. If you draw a right triangle from the origin to the point (r, θ), the horizontal leg has length r·cos(θ) and the vertical leg has length r·sin(θ). Hence: x = r·cos(θ) and y = r·sin(θ). The angle θ may be supplied in degrees or radians; when using degrees, multiply by π/180 before applying the trigonometric functions. Negative values of r are valid and reflect the point through the origin, giving x = |r|·cos(θ + π) and y = |r|·sin(θ + π), which the formulas handle automatically.
Practical applications of this conversion appear throughout science and engineering. In robotics, joint angles and arm lengths (polar) must be converted to workspace positions (Cartesian) for motion planning. In electrical engineering, phasors expressed in polar form (magnitude and phase angle) are converted to rectangular form for addition and subtraction of complex numbers. In computer graphics, rotating sprites or rendering radial gradients requires moving fluidly between both coordinate systems. Surveyors use polar measurements (bearing and distance) and convert them to grid coordinates for mapping.
Worked example
Problem: A point is located at polar coordinates r = 5 and θ = 53.13°. Find its Cartesian coordinates.
Step 1 — Convert angle to radians:
θ (rad) = 53.13° × π/180 = 0.9273 rad
Step 2 — Compute x:
x = r·cos(θ) = 5 × cos(0.9273) = 5 × 0.6000 = 3.000
Step 3 — Compute y:
y = r·sin(θ) = 5 × sin(0.9273) = 5 × 0.8000 = 4.000
Result: The Cartesian coordinates are approximately (3.000, 4.000). This is the classic 3-4-5 right triangle, confirming that r = √(3² + 4²) = 5. ✓
Limitations & notes
The formulas x = r·cos(θ) and y = r·sin(θ) are exact for all real r and θ, but several practical considerations apply. First, floating-point arithmetic introduces rounding errors for angles near 90°, 180°, 270°, or 0°, where one of the trigonometric functions is theoretically zero but may return a very small non-zero number (e.g., 6.12 × 10⁻¹⁷ instead of 0). Second, this calculator performs 2D conversion only; for 3D spherical or cylindrical coordinates, separate formulas involving a polar angle φ and azimuthal angle are needed. Third, the conversion is not unique in the reverse direction: many (r, θ) pairs map to the same (x, y) point — for example, (r, θ) and (r, θ + 2π) are identical — so if you need the inverse (Cartesian to polar), use the atan2 function to recover the correct quadrant. Finally, extremely large values of r combined with very small angles can cause precision loss due to catastrophic cancellation in floating-point subtraction.
Frequently asked questions
What is the difference between polar and Cartesian coordinates?
Cartesian coordinates locate a point using two perpendicular distances (x and y) from a fixed origin along horizontal and vertical axes. Polar coordinates locate the same point using a radial distance r from the origin and an angle θ from a reference direction. Polar form is often more natural for circular or radial phenomena, while Cartesian form is more convenient for linear algebra and most computational algorithms.
Should I enter the angle in degrees or radians?
This calculator accepts both — simply select the correct unit from the dropdown before computing. Mathematically, the formulas require radians, so degree inputs are automatically multiplied by π/180 internally. When in doubt, radians are the standard unit in calculus and physics; degrees are more common in navigation and everyday geometry.
Can r be negative in polar coordinates?
Yes. A negative r reflects the point through the origin, equivalent to adding 180° (or π radians) to the angle. For example, the polar point (−3, 45°) gives the same Cartesian location as (3, 225°). The conversion formulas x = r·cos(θ) and y = r·sin(θ) handle negative r correctly without any adjustment.
How do I convert Cartesian coordinates back to polar?
The inverse conversion uses r = √(x² + y²) for the radial distance, and θ = atan2(y, x) for the angle, where atan2 is the four-quadrant arctangent function available in most programming languages and calculators. Using the standard arctan(y/x) alone is insufficient because it cannot distinguish between opposite quadrants, such as (1, 1) and (−1, −1).
Is this converter useful for complex numbers?
Absolutely. A complex number written in polar form as r·e^(iθ) = r(cos θ + i·sin θ) has real part x = r·cos θ and imaginary part y = r·sin θ, which are exactly the Cartesian conversion formulas. This connection, known as Euler's formula, makes the polar-to-Cartesian conversion indispensable in electrical engineering, control theory, and quantum mechanics when working with phasors and complex exponentials.
Last updated: 2025-01-15 · Formula verified against primary sources.