Polar to Rectangular Coordinates Calculator
Convert a polar point (r, θ) into Cartesian coordinates (x, y) with live validation and a genuine Excel workbook export.
Polar coordinates
y = r × sin(θ)
Rectangular coordinates
Calculation details
| Output | Applied formula | Value |
|---|---|---|
| x coordinate | x = 5 × cos(60°) | 2.5 |
| y coordinate | y = 5 × sin(60°) | 4.330127 |
The table uses the same unrounded model values as the result cards and Excel workbook. Values shown on screen are rounded to six decimal places.
How to use the polar to rectangular coordinates calculator
What this calculator does
This calculator converts one point from polar coordinates, written as (r, θ), into rectangular or Cartesian coordinates, written as (x, y). It applies the exact coordinate identities x = r cos(θ) and y = r sin(θ). The result identifies the same point on a two-dimensional plane; only the coordinate system changes. The tool does not graph an entire polar equation or infer a curve from a single point.
When to use it
Use the converter when checking trigonometry homework, translating a magnitude-and-direction measurement into horizontal and vertical components, preparing coordinate data for a spreadsheet, or verifying an intermediate step in analytic geometry. The underlying relationship is described in the OpenStax discussion of polar coordinates.
How to calculate
- The calculator opens with the demonstration point r = 5 and θ = 60°. Its result and a validated example XLSX workbook are ready immediately.
- Replace Radius (r) with the point's non-negative distance from the origin.
- Replace Angle (θ) with its direction measured from the positive x-axis.
- Choose Angle unit as degrees or radians. Switching units converts the current angle value.
- Read Cartesian point (x, y), x coordinate, and y coordinate. The calculation details table shows the two substitutions.
- Select Download Excel to export the current typed inputs and unrounded outputs. Reset clears the demonstration values and results; export remains disabled until a complete valid state is entered again.
Input guide
Radius (r) is required and represents distance from the origin in generic linear units. Enter a plain decimal from 0 through 1e150, using a period for decimals; for example, 5. A larger radius scales both rectangular components proportionally. Negative radius values are rejected because this calculator follows the non-negative-radius convention. Do not enter unit symbols, decimal commas, or scientific notation.
Angle (θ) is required and represents direction from the positive x-axis. It accepts a signed plain decimal, such as 60 degrees or 1.0471975512 radians. Positive angles rotate counterclockwise and negative angles rotate clockwise. Angles outside one revolution are valid because trigonometric functions are periodic. A common mistake is entering a degree value while the unit is set to radians.
Angle unit is required and determines how θ is interpreted. Choose Degrees (°) for values such as 60 or Radians (rad) for values such as π/3 entered numerically as approximately 1.0471975512. The selector converts a valid current angle during a unit change, so a round trip preserves the represented direction apart from display precision.
Output guide
Cartesian point (x, y) is the combined rectangular coordinate pair. x coordinate is the horizontal component r cos(θ), while y coordinate is the vertical component r sin(θ). All three are exact coordinate conversions before display rounding, not forecasts or recommendations. A zero component means the point lies on an axis; negative x or y values place the point to the left of or below the origin. The summary pills repeat the current radius, angle, and point for quick scanning.
The Calculation details table has three columns: Output names x or y, Applied formula shows the current substitution, and Value displays the corresponding component. Both rows are driven by the same canonical calculation used by the cards and workbook.
Worked example
With the startup values r = 5 and θ = 60°, convert the angle to trigonometric values: cos(60°) = 0.5 and sin(60°) ≈ 0.8660254038. Then x = 5 × 0.5 = 2.5, and y = 5 × 0.8660254038 ≈ 4.3301270189. Rounded to six decimal places, the displayed Cartesian point is (2.5, 4.330127). The identity x² + y² = r² provides a useful independent check.
Learn more
For the geometric meaning of sine and cosine as horizontal and vertical coordinates, review the OpenStax unit-circle explanation. For a compact reference covering polar notation and coordinate transformations, see Wolfram MathWorld's polar coordinates article.
Formula, precision, and edge cases
The calculator converts degrees to radians internally because JavaScript trigonometric functions use radians. It preserves full floating-point precision in the model and workbook, while the interface rounds ordinary values to six decimal places. Tiny values caused by floating-point approximation, such as sin(π), are normalized to zero when they are negligible relative to the radius. When r = 0, every angle maps to the origin (0, 0). Very large finite inputs are limited to ranges that keep all derived results finite.
Common interpretation mistakes
- Swapping x and y: cosine produces x, while sine produces y.
- Mixing degrees and radians: 60 radians is not the same direction as 60 degrees.
- Rounding the angle too early: retain adequate precision when entering a radian approximation.
- Treating equivalent angles as different points: θ, θ + 2π, and θ + 360° describe the same direction.