CALCULATE WITH CONFIDENCE

Angles & Rotation

Rotation Calculator

Rotate a point about any chosen center in the Cartesian plane.

Angles & Rotation

Enter values

Private calculation in your browser
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example

Use finite decimals in the stated units. Separate vector components with commas or spaces. Inputs stay on this device.

Result

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Enter valid values to see the result.

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Understand the subject

What is Rotation?

A rotation turns every point around a chosen center by the same directed angle while preserving distance.

Use it to map a point exactly while distinguishing clockwise from counterclockwise turns.

The relationship

Write the rule before calculating

See the structure

What the calculation is doing

Worked interpretation

Read the result in context

Rotating (1,0) by 90° around the origin gives (0,1).

Interpret with care

Important boundary

Positive Cartesian rotation is counterclockwise; changing the center changes every output coordinate.

Use the Mathematics collection to move between connected concepts without duplicating the calculation.

Quick guide

How to use this calculator

  1. Choose the calculation mode when the page supports more than one direction.
  2. Enter the known measurements using the labels and angle units shown.
  3. Read the primary result first, then use the supporting rows to verify the convention or relationship.

Calculation method

Apply the selected angle relationship

x′=cx+(x−cx)cosθ−(y−cy)sinθ; y′=cy+(x−cx)sinθ+(y−cy)cosθ.

The calculator validates geometric domains and reports degenerate vectors, impossible measurements, or unsupported ranges explicitly.

Worked example

Worked example

Rotating (1,0) by 90° around the origin gives (0,1).

x′=cx+(x−cx)cosθ−(y−cy)sinθ; y′=cy+(x−cx)sinθ+(y−cy)cosθ.

Supported inputs

Precision and limits

Angle conventions

Positive Cartesian rotations are counterclockwise. North-referenced azimuths run clockwise. Unoriented line and plane calculators return the smaller acute or right angle; directed vector and dihedral tools can return up to 180°.

Numerical range

Inputs may be zero or have absolute magnitude from 1e-100 through 1e100. General angle magnitudes are limited to 1,000,000,000 degrees so periodic reduction remains meaningful in double-precision arithmetic.

Precision

Calculations use finite double-precision arithmetic and normally display 12 significant digits. Results are numerical unless the page explicitly displays a fixed geometric identity.