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Coordinate Rotation Calculator

Rotate a point in a two-dimensional Cartesian plane without changing its distance from the origin. Enter the original x and y coordinates and a degree angle. Positive angles turn the point counterclockwise; negative angles turn it clockwise. This is an active rotation of the point while the coordinate axes stay fixed. If your problem rotates the axes instead, reverse the sign to match that different convention.

Enter every field, including explicit zeros. Decimal and scientific notation are accepted; commas, fractions and blank fields are rejected. Each input must be finite and no larger than 1 billion in absolute value.

Rotated x0
Rotated y1
Distance from origin1

How to use this calculator

  1. Match each input to its labeled coordinate, coefficient or matrix position.
  2. Check the signs and use one consistent coordinate convention.
  3. Select Calculate, or press Enter, and read the labeled results.

Formula and method

x′ = x cosθ − y sinθ; y′ = x sinθ + y cosθ.

The calculation runs locally in your browser. Changing an input requires selecting Calculate again. If an input or mathematical condition is invalid, the previous result is hidden instead of leaving a stale answer visible.

Worked example

Rotating (1,0) counterclockwise by 90° around the origin gives (0,1), still one unit from the origin.

Limitations and interpretation

Rotation is about the origin only. To rotate around another center, subtract the center coordinates first, rotate, then add them back. This is not a geographic latitude-longitude rotation or a 3D transformation. Very large degree angles may incur extra trigonometric rounding; reduce them modulo 360 when possible.

Results are displayed to approximately 10 significant digits. The underlying calculation uses ordinary floating-point numbers and is not exact symbolic arithmetic. Rounded results can differ slightly from hand calculations. For a measured quantity, choose precision that reflects the measurements rather than all the digits shown. Verify consequential calculations independently.

Frequently asked questions

Can I rotate clockwise?

Yes. Enter a negative angle, such as −90 for a quarter turn clockwise.

Can I enter negative or decimal values?

Yes, provided the values meet the mathematical conditions described above. Type a normal hyphen for a negative sign. Do not type units, commas or a fraction such as 1/2; enter 0.5 instead.

Why did the calculator hide my previous result?

A missing or invalid input, undefined operation or unstable numerical case produces an explanation. Correct the input and calculate again to obtain a result for the current values.

Review notes and reference

Reviewed October 3, 2026. The stated default example and a boundary or algebraic check were tested against expected values. Input validation checks included blank fields, nonnumeric text and values outside the supported range. Wolfram MathWorld formula reference provides mathematical background. This tool is for calculation and education; its output alone does not validate a professional design.

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