Polar to Cartesian Coordinates Calculator
Convert a polar point (r, θ) into its Cartesian coordinates (x, y) with degree or radian input and a calculation-ready Excel workbook.
Polar coordinates
Conversion: x = r × cos(θ) and y = r × sin(θ).
Cartesian result
Calculation breakdown
| Quantity | Operation | Current value |
|---|---|---|
| Cosine component | cos(60°) | 0.5 |
| Sine component | sin(60°) | 0.866025 |
| x coordinate | 5 × 0.5 | 2.5 |
| y coordinate | 5 × 0.866025 | 4.330127 |
| Magnitude check | √(x² + y²) | 5 |
The magnitude check should equal |r| apart from tiny floating-point rounding differences.
How to use the polar to Cartesian calculator
What this calculator does
This calculator converts one point from polar form, written as (r, θ), to Cartesian form, written as (x, y). Polar coordinates describe a point by a signed radius and an angle from the positive x-axis. Cartesian coordinates describe that same point by horizontal and vertical displacement. The calculation is an exact trigonometric identity in the mathematical model; the displayed decimals are rounded for readability. It does not graph an entire polar equation or choose a unique polar representation, because many angle values can identify the same point.
When to use it
Use the converter when you need rectangular components for a vector, when a geometry problem supplies distance and direction, when software expects x-y coordinates, or when you want to check hand calculations involving sine and cosine. For a broader treatment of the coordinate systems and their conversion identities, see the OpenStax section on polar coordinates.
How to calculate
- The calculator opens with a complete demonstration: r = 5, θ = 60°, and a validated Excel export ready immediately.
- Replace Radius (r) with the signed distance you want to convert. Enter a plain decimal using a period, such as 7.25 or -3.
- Replace Angle (θ) with the direction and choose Angle unit. Switching between degrees and radians converts the current angle rather than merely relabeling it.
- Read the ordered pair and the separate x and y cards. Review the breakdown table to see the cosine, sine, multiplication steps, and magnitude check.
- Select Download Excel to create a current-state workbook with Summary, Inputs, and Calculation sheets. Select Reset to clear the demonstration data. Reset may disable Excel export until both required numbers are entered again.
Input guide
Radius (r) is required and accepts a finite real number between – 1 trillion and 1 trillion. Its unit is whichever distance unit your problem uses; the x and y outputs retain that same unit. A value of 5 means the point is five units from the origin. A larger absolute radius moves the point farther from the origin. A negative radius is valid in polar mathematics and places the point in the direction opposite θ. Do not paste unit symbols or scientific notation.
Angle (θ) is required and accepts a finite real number in the selected angle unit. Values outside one full turn are allowed because the calculator normalizes them internally for interpretation. For example, 420° has the same direction as 60°. A common mistake is entering radians while the unit remains degrees. The Angle unit control accepts Degrees (°) or Radians (rad). NIST identifies the radian as the coherent SI unit for plane angle in its SI guidance on plane-angle units.
Output guide
Ordered pair (x, y) is the converted Cartesian point. x coordinate measures horizontal displacement and is driven by r and cos(θ); positive x is right of the origin and negative x is left. y coordinate measures vertical displacement and is driven by r and sin(θ); positive y is above the origin and negative y is below. Values near zero indicate that the point lies on or very close to an axis.
Normalized angle restates the entered direction within one complete turn, in the selected unit. Location reports a quadrant, axis, or origin classification from the calculated x and y signs. Magnitude check calculates √(x² + y²) and should equal |r|; it is a verification identity, not a separate estimate. In the table, Quantity names each intermediate, Operation shows the current formula substitution, and Current value shows the numeric result.
Worked example
With the startup values r = 5 and θ = 60°, cos(60°) = 0.5 and sin(60°) ≈ 0.8660254038. Therefore x = 5 × 0.5 = 2.5, while y = 5 × 0.8660254038 ≈ 4.330127. The first-open Cartesian result is (2.5, 4.330127). The magnitude check gives √(2.5² + 4.330127²) ≈ 5, confirming the conversion. The same component logic is explained in the OpenStax discussion of coordinate systems and vector components.
How the conversion works
The radius acts as the hypotenuse of a right triangle whose legs are x and y. Cosine returns the horizontal fraction of that radius, and sine returns the vertical fraction. Multiplying those fractions by r produces the Cartesian components. Because sine and cosine repeat every full rotation, θ, θ + 360°, and θ + 2π radians describe the same direction.
Negative radii are also meaningful. A point written with – r at angle θ is equivalent to the point written with positive r at θ + 180° or θ + π radians. The calculator follows the formulas directly, so this reversal appears automatically in the signs of x and y.