GEOMETRYCoordinate GeometryMathematics Calculator
↻

2D Rotation

Rotate a point (x,y) by angle θ about the origin: x' = x cos θ − y sin θ, y' = x sin θ + y cos θ. Positive θ = counterclockwise. For custom center (cₓ,cᵧ): translate, rotate, translate back.

Concept Fundamentals
x' = x cos θ − y sin θ
Origin
(x,y) → (−y,x)
90° CCW
(x,y) → (−x,−y)
180°
R(θ) = [cos −sin; sin cos]
Matrix

Did our AI summary help? Let us know.

90° CCW: (x,y) → (−y,x). Rotation is an isometry (preserves distance). Complex: z·e^(iθ) rotates z by θ.

Key quantities
x' = x cos θ − y sin θ
Origin
Key relation
(x,y) → (−y,x)
90° CCW
Key relation
(x,y) → (−x,−y)
180°
Key relation
R(θ) = [cos −sin; sin cos]
Matrix
Key relation

Ready to run the numbers?

Why: Rotation is fundamental in computer graphics, physics (rigid body motion), and complex numbers (multiply by e^(iθ)). Preserves distances and angles (isometry).

How: About origin: apply rotation matrix. About (cₓ,cᵧ): subtract center, rotate, add center back. x' = cₓ + (x−cₓ)cos θ − (y−cᵧ)sin θ, y' = cᵧ + (x−cₓ)sin θ + (y−cᵧ)cos θ.

90° CCW: (x,y) → (−y,x).Rotation is an isometry (preserves distance).
Sources:Khan Academy — RotationsWolfram MathWorld — Rotation Matrix

Run the calculator when you are ready.

Rotate PointEnter point and angle

Input

Point to Rotate

Rotation

Positive = CCW, Negative = CW

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

↻

x' = x cos θ − y sin θ, y' = x sin θ + y cos θ.

— Rotation Matrix

↺

90° CCW: (x,y) → (−y,x).

— Shortcut

Key Takeaways

  • • About origin: x′=xcostheta−ysinthetax' = x\\cos\\theta - y\\sin\\theta, y′=xsintheta+ycosthetay' = x\\sin\\theta + y\\cos\\theta
  • • About (cₓ, cᵧ): Translate so center is origin, rotate, translate back
  • • Positive θ = counterclockwise; negative θ = clockwise
  • • Rotation is an isometry — preserves distances and angles
  • • 90° CCW: (x,y) → (−y,x); 180°: (x,y) → (−x,−y)

Did You Know?

Rotation Matrix

The 2×2 rotation matrix R(θ) has rows [cos θ, -sin θ] and [sin θ, cos θ].

Complex Numbers

Multiplying by e^(iθ) rotates a complex number by θ. z·e^(iθ) = (x+iy)(cos θ + i sin θ).

Computer Graphics

Rotation is fundamental in 2D/3D graphics. GPU hardware accelerates matrix multiplication.

Physics

Angular displacement, rigid body motion, and orbital mechanics use rotation transformations.

90° Shortcut

90° CCW: swap coordinates and negate first. (x,y) → (-y,x).

180° Shortcut

180°: negate both. (x,y) → (-x,-y).

Understanding the Formulas

About origin (0,0):

x′=xcostheta−ysintheta,quady′=xsintheta+ycosthetax' = x\\cos\\theta - y\\sin\\theta, \\quad y' = x\\sin\\theta + y\\cos\\theta

About center (cₓ, cᵧ): First subtract center, rotate, then add center back:

x′=cx+(x−cx)costheta−(y−cy)sinthetax' = c_x + (x - c_x)\\cos\\theta - (y - c_y)\\sin\\theta
y′=cy+(x−cx)sintheta+(y−cy)costhetay' = c_y + (x - c_x)\\sin\\theta + (y - c_y)\\cos\\theta

Expert Tips

90° CCW

(x,y) → (-y,x). Example: (3,2) → (-2,3).

90° CW

(x,y) → (y,-x). Or use -90° in the formula.

180°

(x,y) → (-x,-y). Just negate both coordinates.

Custom Center

Enable "Use custom center" and enter (cx, cy). The point rotates around that center.

Frequently Asked Questions

What is counterclockwise?

In the standard math coordinate system, counterclockwise is the direction from positive x-axis toward positive y-axis (left-hand rule).

How do I rotate clockwise?

Use a negative angle. -90° gives clockwise 90°.

What about 3D rotation?

3D rotation requires an axis. This calculator handles 2D rotation in the xy-plane (about the z-axis).

Does order matter for multiple rotations?

Yes. Rotating by 90° then 90° is 180°. But rotating A then B is not the same as B then A in general.

What is the rotation matrix?

R(θ) = [[cos θ, -sin θ], [sin θ, cos θ]]. The columns are the images of (1,0) and (0,1).

How do I rotate a shape?

Apply the rotation formula to each vertex of the shape. The shape rotates as a rigid body.

Radians vs degrees?

This calculator uses degrees. To use radians: θ_rad = θ_deg × π/180.

How to Use This Calculator

  1. Enter the point (x, y) and rotation angle θ in degrees (positive = CCW, negative = CW).
  2. Optionally enable "Use custom center" and enter (cₓ, cᵧ) to rotate about a point other than the origin.
  3. Click a sample example to auto-fill and calculate.
  4. Click Calculate to get the rotated coordinates.
  5. Review the visualization and step-by-step solution.

Disclaimer: This calculator uses degrees. For radians, convert first (rad = deg × π/180). Results may have minor rounding.

👈 START HERE
⬅️Jump in and explore the concept!
AI

Related Calculators

Centroid Calculator

Centroid Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Endpoint Calculator

Endpoint Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Gradient Calculator

Gradient Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Vector Calculator

Vector Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Y Intercept Calculator

Y Intercept Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Rise Over Run Calculator

Rise Over Run Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics