Rotation Calculator

A rotation in coordinate geometry is a transformation that moves every point of a figure by a given angle around a fixed center point, called the pivot or center of rotation. The shape and size of the figure are preserved — only its orientation and position change. Rotations are classified as clockwise or counterclockwise.

Using the rotation formulas x' = x·cos(θ) − y·sin(θ) and y' = x·sin(θ) + y·cos(θ), substitute x=3, y=4, θ=60°. That gives x' = 3·cos(60°) − 4·sin(60°) = 3·0.5 − 4·0.866 ≈ −1.964, and y' = 3·sin(60°) + 4·cos(60°) = 3·0.866 + 4·0.5 ≈ 4.598. So the rotated point is approximately (−1.964, 4.598).

To rotate a point (xᵢ, yᵢ) around the origin by angle θ, apply: x' = xᵢ·cos(θ) − yᵢ·sin(θ) and y' = xᵢ·sin(θ) + yᵢ·cos(θ). Make sure your angle is in radians when using trigonometric functions in most programming languages (multiply degrees by π/180 to convert).

To rotate around an arbitrary pivot (cx, cy), first translate the point so the pivot becomes the origin: x_t = xᵢ − cx, y_t = yᵢ − cy. Then apply the standard rotation: x'_t = x_t·cos(θ) − y_t·sin(θ), y'_t = x_t·sin(θ) + y_t·cos(θ). Finally, translate back: x' = x'_t + cx, y' = y'_t + cy.

The 2D rotation matrix for an angle θ is: R(θ) = [[cos θ, −sin θ], [sin θ, cos θ]]. Multiplying this matrix by the column vector [x, y]ᵀ gives the rotated coordinates [x', y']ᵀ. This formalism is especially useful when rotating multiple points or composing several transformations.

Counterclockwise rotation is the standard positive direction in mathematics and follows the standard rotation formulas. Clockwise rotation is the negative direction and is equivalent to rotating by −θ counterclockwise. In the clockwise case the formula becomes: x' = x·cos(θ) + y·sin(θ) and y' = −x·sin(θ) + y·cos(θ).

No. Rotation is an isometric transformation, meaning it preserves distances. The distance of a point from the center of rotation (its radius) remains exactly the same before and after rotation. Only the angle changes. You can verify this by computing √(x'² + y'²) and comparing it to √(x² + y²).

For 90° counterclockwise: (x, y) → (−y, x). For 180°: (x, y) → (−x, −y). For 270° counterclockwise (or 90° clockwise): (x, y) → (y, −x). These are special cases where the cosine and sine values simplify to 0 and ±1, making the arithmetic very straightforward.