Rotation Calculator

How to use this 2D rotation calculator

What this calculator does

This calculator applies a two-dimensional rigid rotation to one through ten Cartesian points. It returns each point's final x- and y-coordinate after turning it through a chosen angle around a chosen pivot. It also reports the sine and cosine used by the rotation matrix, compares the centroid before and after the transformation, checks that every point remains the same distance from the pivot, draws a coordinate map when at least three complete points are available, and prepares a real Excel workbook from the current model. It does not infer an angle from two shapes, solve a three-dimensional rotation, or choose a pivot for you.

When to use it

Use it to check coordinate-geometry homework, transform vertices for a drawing or game scene, rotate survey or CAD-style point sets in a plane, or verify a hand-built rotation matrix. The equal-scale plot is especially useful when you want to confirm that a polygon's shape and distance from the pivot are preserved.

How to calculate

1. The calculator opens with a complete three-point demonstration and an immediately available example XLSX export. Replace the sample values with your own.
2. Enter the Rotation angle, then choose Angle unit. Positive values mean counterclockwise rotation and negative values mean clockwise rotation. Changing the unit converts the current angle rather than merely relabeling it.
3. Select the Number of points, enter Pivot x and Pivot y, then complete the x and y fields for every displayed point.
4. Read the live result cards, coordinate table, and point map. Select Download Excel to export the current validated values. Select Reset to clear the demonstration and all calculated content; the export button remains unavailable until a complete valid state is entered again.

Input guide

Rotation angle is a required finite decimal. It accepts an optional leading plus or minus sign and a period as the decimal separator, such as 60, -90, or 1.5708. Commas, scientific notation, and unit letters are rejected to prevent ambiguous interpretation. A larger positive angle rotates farther counterclockwise; a negative value reverses direction. Angles outside one full turn are valid and are normalized only for the summary.

Angle unit is required and offers Degrees or Radians. The sample uses 60 degrees. One full turn equals 360 degrees or 2π radians; the OpenStax explanation of angles and radian measure provides the underlying unit relationship. A common mistake is entering a radian value while Degrees remains selected.

Number of points is required and ranges from 1 to 10. Increasing it adds coordinate pairs; reducing it removes the extra point rows from the current model. The sample uses three points so the original and rotated triangle can be compared visually.

Pivot x and Pivot y are required finite Cartesian coordinates for the center of rotation. The sample pivot is (0, 0), the origin. Any point equal to the pivot remains unchanged. Moving the pivot changes every translated offset used by the matrix, so the same angle can produce very different final coordinates.

Each Point n x and Point n y field is required for its displayed point and uses the same strict decimal format. Negative and fractional coordinates are valid. The startup points are (3, 4), (-2, 1), and (5, -1). Leaving either coordinate empty makes the complete calculation invalid; do not substitute a blank with zero unless zero is genuinely the intended coordinate.

Output guide

Point 1 final coordinates is the primary transformed pair. Direction describes the sign of the entered angle, while Normalized angle reports the equivalent position within a 0° to less-than-360° turn. cos θ and sin θ are the exact matrix coefficients at floating-point precision. Centroid before and Centroid after are the arithmetic means of all point coordinates; the centroid rotates by the same rule as the individual points.

The summary pills repeat the current angle, point count, pivot, and Max radius error. Radius error is the absolute difference between each point's distance from the pivot before and after rotation. An ideal rotation preserves that distance exactly, so zero or a tiny rounding-level number is expected. A large value would signal an implementation or data problem.

The Coordinate details table lists Point, Initial x, Initial y, Final x, Final y, Radius before, Radius after, and Radius error. These values are estimates only in the sense that irrational trigonometric results must be displayed as decimals; the underlying transformation is an exact geometric identity. The Original points, Rotated points, and Pivot plot series use the same canonical values as the table.

Worked example

For Point 1 = (3, 4), pivot = (0, 0), and θ = 60°, cos θ = 0.5 and sin θ ≈ 0.866025. The final x-coordinate is 0 + (3 - 0) × 0.5 - (4 - 0) × 0.866025 = -1.964102. The final y-coordinate is 0 + (3 - 0) × 0.866025 + (4 - 0) × 0.5 = 4.598076. Therefore the first-open primary result is (-1.964102, 4.598076). Its radius remains 5 because √(3² + 4²) = 5 and the rotated point has the same distance from the pivot.