Calculator Online

Home AI Chat AI Math Solver

NEW

Tools ▾ Category ▾

Pricing Sign In

Calculator Online

Your Result is copied!

Rotation Calculator

Enter the coordinates of the point you want to rotate, along with the center of rotation and the angle (clockwise or anticlockwise). The calculator will then compute the new position of the rotated point based on the given parameters.

Point to Rotate

Angle

Unit

Point to Rotate Around

degrees

Related

Our free online rotation calculator allows you to rotate any point around a given center point, either clockwise or counterclockwise. You can enter angles in degrees or radians, and it will show the step-by-step calculation for the new coordinates using rotation formulas and matrices.

What is Rotation?

In geometry, rotation is a fundamental transformation. Simply put,

"Rotation is the turning of a shape or object around a fixed point."

For example, the Earth rotates around its axis daily. When an object rotates, it can experience various transformations such as:

Rotation
Reflection
Translation
Scaling (Resizing)

Here, we focus specifically on the rotation of points and objects around a fixed center.

Rotation Formula

The new coordinates $(X, Y)$ of a point $(x, y)$ after rotation by an angle $\theta$ around the origin are given by:

X = x cos(θ) - y sin(θ)
Y = x sin(θ) + y cos(θ)

Where:

X, Y: the rotated point coordinates
θ: rotation angle in degrees or radians

Clockwise vs Counterclockwise Rotation

Rotation can occur in either direction:

Clockwise (CW): represented by a negative angle
Counterclockwise (CCW): represented by a positive angle

Standard rotations include 90°, 180°, and 270°, with resulting coordinates:

Rotation	Original Point (x, y)	Rotated Point
90° Clockwise	(x, y)	(y, -x)
90° Counterclockwise	(x, y)	(-y, x)
180° (Both directions)	(x, y)	(-x, -y)
270° Clockwise	(x, y)	(-y, x)
270° Counterclockwise	(x, y)	(y, -x)

Rotation Matrix

In the Cartesian plane, rotation can also be expressed using a matrix. For a counterclockwise rotation by $\theta$:

$$ R = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} $$

Multiplying this matrix with a point vector rotates it around the origin.

Rotational Symmetry

Many shapes, like squares, triangles, and circles, have rotational symmetry. A shape shows rotational symmetry if it looks identical after rotation by certain angles.

Rotational Symmetry

Real-Life Examples

Ceiling fans spin around their axis.
Doors rotate about their hinges.

Rotational Example

How the Rotation Calculator Works

Input:

Provide the point coordinates.
Choose the rotation angle and unit (degrees or radians).
Select the rotation direction: clockwise or counterclockwise.
Press "Calculate".

Output:

New coordinates calculated using the rotation formula.
Results derived from the rotation matrix method.

FAQs

Is rotation a transformation?

Yes, it is a type of geometric transformation that moves points around a fixed center.

What does rotation mean in math?

It is the movement of a point or object around a fixed center or axis.

Can rotations happen in one direction only?

No, rotation can be clockwise or counterclockwise.

Are rotation and revolution the same?

One full revolution equals 360°, which is equivalent to a complete rotation.

Difference between point of rotation and angle of rotation?

The point of rotation is the center of rotation, and the angle of rotation measures how much the object rotates.

Conclusion

The rotation calculator efficiently computes the new positions of points after rotation. This is helpful for geometry, physics, engineering, and everyday objects like fans, doors, and wheels.

References

From Wikipedia: Rotation, Mathsisfun: Rotation

Related

Calculator Online

Get the ease of calculating anything from the source of calculator online

Links

Home Conversion Calculator About Calculator Online Blog Hire Us Knowledge Base Sitemap Sitemap Two

Support

Email us at