Rotation Calculator — 2D Point Rotation About the Origin
Enter a point (x, y) and a rotation angle to find the new coordinates after rotating about the origin — with the distance from origin and angles before and after.
°
x·cos(θ) − y·sin(θ)
- 1
cos(θ)
cos(90°) = 0 - 2
sin(θ)
sin(90°) = 1 - 3
x · cos(θ)
3 × 0 = 0 - 4
x′ = x·cos(θ) − y·sin(θ)
0 − 0 × 1 = 0
How does this calculator work?
Rotate point (x, y) by angle θ: x′ = x·cos(θ) − y·sin(θ), y′ = x·sin(θ) + y·cos(θ). Positive θ = counter-clockwise. Distance from origin is preserved. For (3, 0) rotated 90°: x′ = 0, y′ = 3.
Formula
How this is calculated
A 2D rotation by angle θ about the origin transforms any point (x, y) to a new point (x′, y′) using the rotation matrix. The formulas are x′ = x·cos(θ) − y·sin(θ) and y′ = x·sin(θ) + y·cos(θ). By convention, positive angles rotate counter-clockwise and negative angles rotate clockwise.
Several geometric properties are preserved under rotation: the distance from the origin (r = √(x² + y²)) is unchanged — rotation is a rigid, length-preserving transformation (an isometry). The angle of the point from the positive x-axis increases by exactly θ. So if the original point is at angle φ, the rotated point is at angle φ + θ.
These formulas are the building blocks of computer graphics, robotics kinematics and game engines. For rotation about an arbitrary point (a, b), translate the point by (−a, −b) to shift that point to the origin, apply the rotation, then translate back by (a, b). This calculator handles the standard origin case; the VectorDiagram below shows the original and rotated vectors from the origin.
Frequently asked questions
Rotation about the origin is an isometry — it preserves distances. The point stays exactly the same distance from the origin: r = √(x² + y²) = √(x′² + y′²). Only the direction changes, not the magnitude.
Translate so the centre of rotation is at the origin: subtract (cx, cy) from your point, apply the rotation formulas, then add (cx, cy) back. For example to rotate (5, 3) around (2, 2) by 90°: shift to (3, 1), rotate to (−1, 3), shift back to (1, 5).
The 2×2 rotation matrix [[cos θ, −sin θ],[sin θ, cos θ]] encodes the same formulas: multiplying it by the column vector [x; y] gives [x′; y′]. Composing two rotations means multiplying their matrices — rotation matrices form the special orthogonal group SO(2).
Also known as
TG we-Calculate Editorial Team. (2026). Rotation Calculator — 2D Point Rotation About the Origin [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/rotation-calculator
TG we-Calculate Editorial Team. "Rotation Calculator — 2D Point Rotation About the Origin." TG we-Calculate. 2026. https://we-calculate.com/calculator/rotation-calculator.
TG we-Calculate Editorial Team, "Rotation Calculator — 2D Point Rotation About the Origin," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/rotation-calculator
@misc{wecalculate_rotation_calculator, title = {Rotation Calculator — 2D Point Rotation About the Origin}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/rotation-calculator}}, year = {2026}, note = {TG we-Calculate} }
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