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DIY Cost Tools

DIY Cost Tools

Polar to Cartesian Calculator

Convert a polar point to rectangular coordinates. The radius is the nonnegative distance from the origin. The angle is measured counterclockwise from the positive x-axis; a negative angle rotates clockwise. Angles outside one turn represent equivalent directions.

Enter every field in the stated units. Decimal and scientific notation are accepted for non-integer fields; blanks, commas, fractions and non-finite entries are rejected. Zero is accepted only where a field permits it.

x coordinate (units)4.330127019
y coordinate (units)2.5

How to use this calculator

  1. Identify the quantities required by the model and use the field units.
  2. Replace the example values, then select Calculate or press Enter.
  3. Read the labeled outputs and check the interpretation below.

Formula and interpretation

x = r cos θ; y = r sin θ, with degrees converted to radians internally.

Calculations run in your browser. The previous results are hidden when an input changes or validation fails. This prevents an old answer from being mistaken for the new calculation. Results are displayed to approximately ten significant digits using floating-point arithmetic; that precision does not imply equal precision in measured inputs.

Worked example

Radius r (units) = 5; Angle θ (degrees) = 30. x coordinate (units) = 4.330127019; y coordinate (units) = 2.5.

Substitute the example inputs into the stated formula in the same order as the labeled fields. These are the default values shown above, so the explanation and the working interface describe the same case.

Limitations

Uses the ordinary two-dimensional convention. It does not convert geographic latitude and longitude. Floating-point trigonometry can leave a tiny residual where theory gives exactly zero.

This is an educational mathematical model. Check the assumptions and units for your situation. Geometry results do not establish material strength, and probability outputs do not predict an individual outcome.

Frequently asked questions

Can the angle exceed 360°?

Yes. Additional turns return the same point up to numerical rounding.

Why does the calculator reject my entry?

Each field shows its supported range. Integer fields reject fractions; mathematical conditions such as a zero divisor or invalid shape produce a separate explanation. Correct the indicated field or condition and calculate again.

How many digits should I use?

Use enough digits to retain your input accuracy. Round the final answer to the precision justified by the data; tiny residuals around theoretical zero can result from floating-point arithmetic.

Review notes and reference

Reviewed October 3, 2026. The calculator script was tested against the default example and separate boundary cases. Checks covered blank, text, comma-formatted, non-finite and out-of-range entries, plus recovery after errors. Background reference for the mathematical method. Explanations and worked examples are original.

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