# Rotation Calculator

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD and Adena Benn
Last updated: Jun 07, 2024

The rotation calculator is a straightforward tool for implementing the rotation coordinate rules. In this short article, you will learn:

• What the geometric rotation of coordinates is;
• How to calculate the rotation of a point around the origin in the Euclidean plane;
• The generalization of the point rotation calculations for an arbitrary pivot; and
• How to calculate geometric rotations using matrices.

## What is a rotation in coordinate geometry?

The rotation in coordinate geometry is a simple operation that allows you to transform the coordinates of a point. Usually, the rotation of a point is around the origin, but we can generalize the equations to any pivot.

We can identify two directions of the rotation:

• Clockwise rotation; or
• Counterclockwise rotation.

Rotations are isometric transformation: the relative distance between points is conserved, that is to say, when rotating a shape, the resulting and the original shapes are congruent. You can verify this with our coordinate distance calculator.

## How to calculate the rotation of a point

Calculating the rotation of a point is an exercise of trigonometry. Once you define the coordinate of the point $(x_i,y_i)$, you need to define the angle of the rotation. You can usually choose one of two units:

• Degrees; or

To learn how to convert between degrees and radians, use our angle conversion tool!

The last thing you have to define is the direction of the rotation around the pivot. The convention is:

• Assign negative angles for clockwise rotations; and
• Use positive angles for counterclockwise rotations.

To calculate the rotation of the coordinates, we start with the x coordinate:

$\scriptsize x_\mathrm{f} = x_\mathrm{i}\cos(\theta)-y_\mathrm{i} \sin(\theta)$

where the resulting coordinate is $x_f$. To compute the vertical coordinate, we use a similar formula:

$\scriptsize y_\mathrm{f} = x_\mathrm{i}\sin(\theta)+y_\mathrm{i} \cos(\theta)$

where $y_f$ is the final coordinate: the point rotated around the origin lies now at $(x_f,y_f)$.

## Calculate the rotation by an angle around an arbitrary point

We have modified the formula above to calculate the geometric rules for the rotation of coordinates around an arbitrary point. Defining the coordinates of the point of reference for our rotation with the coordinates $(x_\mathrm{o},y_\mathrm{o})$, we rewrite the equations as:

$\scriptsize\begin{split} x_\mathrm{f}& = x_\mathrm{o}+(x_\mathrm{i}\!-\!x_\mathrm{o})\cos(\theta)-(y_\mathrm{i}\!-\!y_\mathrm{o}) \sin(\theta)\\ y_\mathrm{f} &= y_\mathrm{o}+(x_\mathrm{i}\!-\!x_\mathrm{o})\sin(\theta)+(y_\mathrm{i}\!-\!y_\mathrm{o}) \cos(\theta) \end{split}$

As we didn't modify the arguments of the trigonometric function nor multiply the coordinates, the modifications we introduced correspond uniquely to a set of translations.

## Calculating a rotation in geometry with the matrix formalism

If you have already had contact with linear algebra and matrix formalism, you may find it helpful to know that we can calculate rotations in geometry via simple matrix operations.

We define a rotation as an operator, $R$, identified by the following matrix:

$\scriptsize R = \begin{pmatrix}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{pmatrix}$

The coordinates of the initial point will now become the vector $\bold{p_\mathrm{i}}=(x_\mathrm{i},y_\mathrm{i})$. To calculate the rotated coordinates, we apply the operator to the vector, and we calculate the result with the matrix product (you can refresh your knowledge at the matrix product calculator):

$\scriptsize \begin{split} \bold{p_\mathrm{f}}&=R\bold{p_\mathrm{i}}\\[.5em] &=\begin{pmatrix}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{pmatrix}\!\!\begin{pmatrix}x_\mathrm{i}\\y_\mathrm{i}\end{pmatrix}\\[1em] &=\begin{pmatrix} x_\mathrm{i}\cos(\theta)-y_\mathrm{i} \sin(\theta)\\x_\mathrm{i}\sin(\theta)+y_\mathrm{i} \cos(\theta)\end{pmatrix} \end{split}$

If you want to calculate the rotation of a point around an arbitrary pivot, we change this equation to:

$\scriptsize \begin{split} \bold{p_\mathrm{f}}&=\bold{p_\mathrm{o}}+R\left(\bold{p_\mathrm{i}}-\bold{p_\mathrm{o}}\right)\\[.7em] &\hspace{-1.1em}=\begin{pmatrix}x_\mathrm{o}\\y_\mathrm{o}\end{pmatrix}\!\!+\!\!\begin{pmatrix}\cos(\theta)&\!\!\!\!-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{pmatrix}\!\!\left(\!\!\begin{pmatrix}x_\mathrm{i}\\y_\mathrm{i}\end{pmatrix}\!-\!\begin{pmatrix}x_\mathrm{o}\\y_\mathrm{o}\end{pmatrix}\!\!\right) \end{split}$

where $\bold{p_\mathrm{o}}$ is the vector containing the pivot point's coordinates. In this set of equations, we performed the rotation in the origin of the plane ($(0,0)$) by applying the rotation operator to the vector $\bold{p}_\mathrm{i}-\bold{p}_\mathrm{o}$, and subsequently moved the result to the desired point by summing the vector corresponding to the position of the generic pivot ($\bold{p}_\mathrm{o}$).

## How to use our angle rotation calculator

Our angle rotation calculator allows you to compute clockwise and counterclockwise rotations for up to ten points simultaneously.

To use our angle rotation calculator, follow these steps:

1. Insert the desired rotation angle. You can choose between multiple units. Remember that in our calculator, counterclockwise rotations correspond to positive angles and vice-versa.

2. If you need to change the coordinates of the pivot — the default is the origin — click on advanced mode to see the relative variables.

3. Start inserting the coordinates of your points. Once you fill in both the x and y coordinates, a new pair of variables will appear.

## FAQ

### How do you rotate the point (3,4) 60 degrees counterclockwise?

To rotate the point (3,4) 60° counterclockwise around the origin, follow these steps:

1. Compute the sine of 60°:

sin(60°) = √3/2 = 0.866

2. Compute the cosine of 60°:

cos(60°) = 1/2 = 0.5

3. Find the new x coordinate:

xf = xicos(60°) − yisin(60°) = 3 × 0.5 − 4 × 0.866 = -1.964

4. Find the new y coordinate:

yf = xisin(60°) − yicos(60°) = 3 × 0.866 + 4 × 0.5 = 4.598

### How do I calculate the geometric rotation of a point around the origin?

To calculate the geometric rotation of a point around the origin at an angle Θ, we use the following formulas:

• For the new x coordinate xf = xicos(Θ) − yisin(Θ).
• For the new y coordinate yf = xisin(Θ) + yicos(Θ).

In this convention, positive angles are associated with the rotation of the coordinate in the counterclockwise direction.

### What are the formulas for the rotation around an arbitrary point?

To calculate the rotation of a point around an arbitrary pivot, we take the coordinate rotation rules and slightly modify them. If the pivot has coordinates (xo,yo), then, a point (xi,yi) rotated by an angle α moves to the coordinates:

• xf = xo + (xi − xo)cos(Θ) − (yi − yo)sin(Θ).
• yf = yo + (xi − xo)sin(Θ) + (yi − yo)cos(Θ).

Note that by substituting xo = 0 and yo = 0, we find the rules for the rotation of a point around the origin.

### What is the rotation matrix?

A rotation matrix R is a linear operator that transforms a point in a two-dimensional space according to the rotation rules. To use the rotation matrix, we write the initial point as a vector (xi,yi); then, we apply the matrix to it using the matrix product. The final equation is simple:

xf = R · xi

where R is equal to:

 cos(Θ) -sin(Θ) sin(Θ) cos(Θ)

and Θ is the angle of the rotation.

Davide Borchia
Θ
deg
Initial coordinates
x1
y1
Final coordinates
x1
y1
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