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Rotation Calculator

Rotate a point around the origin by a specified angle in the coordinate plane.

Reviewed by Christopher FloiedPublished Updated

This free online rotation calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

x coordinate of the point to rotate

y coordinate of the point to rotate

Counterclockwise rotation angle in degrees

Results

Rotated x Coordinate

-4

Rotated y Coordinate

3

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Rotation Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Rotation Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Rotation Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Rotation Calculator

Rotation is a geometric transformation that turns every point of a figure around a fixed center point by a given angle. When the center is the origin, the rotation can be expressed using the rotation matrix, which applies cosine and sine functions to map old coordinates to new ones. A positive angle represents counterclockwise rotation, while a negative angle represents clockwise rotation. Rotations preserve distances and angles, making them rigid transformations — the shape and size of the figure remain unchanged. This property is why rotations are classified as isometries. In computer graphics, rotations are used constantly for animating objects, changing viewpoints, and constructing complex scenes. Robotics relies on rotation calculations for joint movements and orientation control. Game development uses rotations for character movement, camera control, and physics simulations. In physics, rotational mechanics governs everything from spinning tops to planetary orbits. This calculator applies the two-dimensional rotation matrix to transform a point's coordinates through any angle around the origin.

The Math Behind It

The 2D rotation matrix R(θ) = [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]] transforms a point (x, y) to (x cos(θ) - y sin(θ), x sin(θ) + y cos(θ)). This matrix has determinant 1 and is orthogonal, meaning R⁻¹ = Rᵀ = R(-θ). The rotation matrix preserves the dot product: the angle between any two vectors is unchanged, as is the length of any vector. For rotation about a point (cx, cy) other than the origin, first translate by (-cx, -cy), rotate, then translate back by (cx, cy). Composing two rotations by angles α and β gives a rotation by α + β, since R(α)R(β) = R(α + β). This is a consequence of the angle addition formulas for sine and cosine. In three dimensions, rotations become more complex: they are described by 3×3 orthogonal matrices with determinant 1, or equivalently by quaternions, which avoid the gimbal lock problem that arises with Euler angles. The group of all 2D rotations forms the special orthogonal group SO(2), which is isomorphic to the unit circle. Every rotation can be decomposed into two reflections, a result known as the Cartan-Dieudonne theorem.

Formula Reference

Rotation Matrix

(x', y') = (x·cos(θ) - y·sin(θ), x·sin(θ) + y·cos(θ))

Variables: x, y = original coordinates; θ = rotation angle (counterclockwise); x', y' = rotated coordinates

Worked Examples

Example 1: Rotate 90° counterclockwise

Rotate the point (3, 4) by 90° counterclockwise around the origin.

Step 1:θ = 90° = π/2 radians
Step 2:cos(90°) = 0, sin(90°) = 1
Step 3:x' = 3(0) - 4(1) = -4
Step 4:y' = 3(1) + 4(0) = 3

The rotated point is (-4, 3).

Example 2: Rotate 45° counterclockwise

Rotate (1, 0) by 45° around the origin.

Step 1:θ = 45° = π/4 radians
Step 2:cos(45°) = √2/2 ≈ 0.7071, sin(45°) = √2/2 ≈ 0.7071
Step 3:x' = 1(0.7071) - 0(0.7071) = 0.7071
Step 4:y' = 1(0.7071) + 0(0.7071) = 0.7071

The rotated point is approximately (0.7071, 0.7071).

Common Mistakes & Tips

  • !Forgetting to convert degrees to radians before applying trigonometric functions.
  • !Using a clockwise convention when the formula assumes counterclockwise — negate the angle for clockwise rotation.
  • !Mixing up the signs in the rotation formula: the top row has a minus sign (-sin), the bottom row does not.
  • !Assuming rotations around an arbitrary center work the same as around the origin — translation is needed first.

Related Concepts

Polar Coordinates

Rotations simply add to the angular coordinate in polar form.

Cosine Similarity

Rotation preserves angles, which is measured by cosine similarity.

Three-Dimensional Distance

3D rotations extend the concept to three-dimensional space.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Polar Coordinates Calculator

Frequently Asked Questions

How do I rotate clockwise instead of counterclockwise?

Use a negative angle. For example, a 90° clockwise rotation is the same as a -90° (or equivalently 270°) counterclockwise rotation.

How do I rotate around a point other than the origin?

Translate the point so the center of rotation is at the origin, apply the rotation, then translate back. If the center is (cx, cy), subtract (cx, cy), rotate, then add (cx, cy).

Does rotation change the distance from the origin?

No. Rotation preserves distances from the center of rotation. The point moves along a circle centered at the origin with the same radius.

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