Rotation Calculator - 2D Point Rotation by Any Angle

A rotation calculator is a tool that takes a 2D point, an angle, and an optional center of rotation, and returns the new coordinates of the point after it has been rotated in the Cartesian plane. It applies the standard 2D rotation operator from linear algebra so you can rotate a point around the origin or around any other pivot without doing the trig by hand, and it shows the matching 2x2 rotation matrix for the same angle.

• • Coordinate geometry homework: Rotate polygon vertices around a chosen pivot to verify mirror, quarter-turn, or half-turn constructions.
• • Linear algebra and physics problems: Apply the 2D rotation matrix to a column vector and check the result against the operator form R(θ) = [[cos θ, -sin θ], [sin θ, cos θ]].
• • Computer graphics and game dev: Pre-compute the rotated positions of sprites, hitbox corners, or camera anchors before pushing them to a renderer or physics engine.
• • Robotics and kinematics: Convert a point expressed in one local frame to a rotated frame around a base joint without re-deriving cos and sin by hand.

Rotation is an isometry, which means the distance from the rotated point to the pivot stays the same before and after the operation, so rotating a whole set of vertices around a shared pivot preserves the figure.

If you need to switch the angle above to the other unit before entering it, the angle converter handles degrees, radians, and gradians in one form.

The calculator translates the point so the pivot sits at the origin, applies the 2D rotation operator, then translates the result back. The same logic works whether the pivot is (0, 0) or any other coordinate pair, and the matrix is the operator form of the rotation in radians.

• xi, yi: Initial point coordinates you want to rotate.
• xo, yo: Pivot (center of rotation). Defaults to (0, 0) for rotation around the origin.
• θ: Rotation angle, positive for counterclockwise and negative for clockwise. Internally always converted to radians.
• cos θ, sin θ: Trig values of the angle, computed once and reused for the coordinates and the matrix.
• xf, yf: Output coordinates of the rotated point.

The order matters: translate by −(xo, yo), rotate by θ, then translate by +(xo, yo). The calculator does this in one pass so the result agrees with the standard matrix formalism.

According to Wolfram MathWorld, the 2D rotation matrix is R(θ) = [[cos θ, −sin θ], [sin θ, cos θ]] and acts on a column vector (x, y) to give the rotated coordinates.

According to Wikipedia (Rotation matrix), rotating a point (xi, yi) by an angle θ around a pivot (xo, yo) gives xf = xo + (xi − xo) cos θ − (yi − yo) sin θ and yf = yo + (xi − xo) sin θ + (yi − yo) cos θ, with positive θ corresponding to a counterclockwise rotation.

When the rotation result is the first step in a circle problem, the arc length calculator turns the same angle into the arc length swept around the pivot, which is r · θ with θ in radians.

Four ideas cover most of what you need to interpret the result: the direction convention, the role of the pivot, the rotation matrix, and the special angles that always give clean results.

If the rotation is part of a circle problem, the central angle calculator turns the same angle into a chord or arc length.

The calculator updates as you type, so you can change any of the six inputs and see the rotated coordinates and matrix in real time without clicking a button first.

1. 1 Choose degrees or radians: Use the Angle Unit selector to tell the calculator whether the angle below is in degrees or radians. Pick radians only if you already have a value in radian form.
2. 2 Enter the rotation angle: Type the angle into the Rotation Angle field. Use a positive value for a counterclockwise rotation and a negative value for a clockwise one.
3. 3 Enter the initial point: Fill in Initial x and Initial y with the coordinates of the point you want to rotate. Decimals are fine; the calculator does not require integer values.
4. 4 Set the pivot: Enter the pivot coordinates. Leave both fields at 0 to rotate around the origin, or set any other point to rotate around that center.
5. 5 Read the rotated coordinates and matrix: The Final x and Final y cards show the rotated point, and the Rotation Matrix card shows the matching 2x2 operator R(θ) for the angle you used.
6. 6 Adjust until the geometry matches: Tweak the angle, the initial point, or the pivot and the calculator re-renders the result. This is the fastest way to line up homework answers, sprite positions, or a kinematic chain.

Worked example: type angle 60, unit degrees, point (3, 4), pivot (0, 0). The calculator returns xf ≈ −1.964102, yf ≈ 4.598076, with matrix [[0.5, −0.866], [0.866, 0.5]]. Change the pivot to (1, 1) and the same angle moves the rotated point to (0.035898, 5.598076).

Once the rotation calculator hands you the new (x, y), the coordinate plane calculator is the easiest way to plot both the original and rotated points on the same grid.

Using a rotation calculator removes the most common sources of sign and unit mistakes and gives you a matrix you can paste into a linear algebra problem.

• • Fewer trig slips: The calculator handles the sin and cos lookup internally, so you no longer have to remember whether the sign goes on the y term or the x term when you rotate around an arbitrary pivot.
• • Degrees and radians in one form: Switch the Angle Unit selector instead of manually converting 60° to π/3 rad. The calculator multiplies by π/180 when the unit is degrees and skips the conversion otherwise.
• • Works for any pivot: Pivot at (0, 0) for the textbook case, or at the center of a triangle, the center of a sprite, or a robot joint origin. The translate–rotate–translate logic is built in.
• • Includes the rotation matrix: Each result ships with the 2x2 operator R(θ). Drop it straight into a linear algebra answer, a graphics shader, or a kinematics equation.
• • Real-time feedback: Every keystroke recomputes xf, yf, cos θ, sin θ, and the matrix, so you can sweep an angle from 0° to 360° and watch the path of the point without rebuilding the input.

After rotating the point, use the 2D distance calculator to confirm that the distance from the pivot is the same as before the rotation.

Three controls drive almost every result, and two limitations keep the output honest about what the math can and cannot do.

• • This tool is 2D only. A 3D turn needs a separate axis-angle or quaternion tool, because a 2D pivot and a 2x2 matrix cannot represent a 3D rotation.
• • The output is a single point, not a polygon. To rotate several vertices, loop the rotation formula over each vertex, or stack R(θ) into a transformation sequence for the whole figure.

When the result is part of a larger geometry problem, the safest verification is to re-run with the opposite angle, or to check the rotated point's distance to the pivot against the original point's distance to the same pivot.

According to Wolfram MathWorld, the 2D rotation matrix satisfies the orthogonality condition R(θ) · R(θ)ᵀ = I, so its inverse equals its transpose, det(R(θ)) = 1, and a clockwise rotation by θ cancels a counterclockwise rotation by the same angle.

Because a rotation traces a circle around the pivot, the circle calculator is the natural next stop for area and circumference of that circle.