Rotation Transformation Calculator

Work out rotation transformation instantly with clear inputs, formula shown and shareable results.

About Rotation Transformation Calculator

Rotating a point counter-clockwise about the origin by θ maps (x, y) to (x·cos θ − y·sin θ, x·sin θ + y·cos θ). Rotation preserves distance from the centre, so the radius is unchanged.

Formula

Rotation about the origin

x' = x·cos θ − y·sin θ, y' = x·sin θ + y·cos θ

Worked example

With the default values below, Rotation Transformation Calculator returns Image point of (-4, 3). Change any input above and every figure here is recalculated in your browser.

Inputs used

Point x: 3
Point y: 4
Rotation angle (counter-clockwise): 90 °

Results

Image point: (-4, 3)
New x: -4.0000
New y: 3.0000
Distance from origin (unchanged): 5.0000

Step by step

1Values used
2Rotation about the origin
3Image point= (-4, 3)
4New x= -4.0000
5New y= 3.0000
6Distance from origin (unchanged)= 5.0000

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Rotating (3, 4) by 90°?

It becomes (−4, 3), and the distance from the origin stays 5.

How do I rotate clockwise?

Enter a negative angle, which is equivalent to using −θ in the formulas.