Rotate a point around a pivot
Enter a point (x, y), choose a pivot (cx, cy), and provide an angle. Use clockwise to rotate with a negative sign on θ.
Formula used
x′ = cx + (Δx·cosθ) − (Δy·sinθ)
y′ = cy + (Δx·sinθ) + (Δy·cosθ)
How to use this calculator
- Enter the point coordinates you want to rotate.
- Set the pivot coordinates, or keep (0,0) for the origin.
- Type the angle and choose degrees or radians.
- Select direction and decimal places, then submit.
- Use CSV or PDF export for your records.
Example data table
| Point (x, y) | Pivot (cx, cy) | Angle | Direction | Rotated (x′, y′) |
|---|---|---|---|---|
| (2, 3) | (0, 0) | 90° | Counterclockwise | (−3, 2) |
| (5, 0) | (1, 1) | 45° | Clockwise | (4.536, −1.121) |
| (−1, 4) | (2, 2) | π/2 rad | Counterclockwise | (0, −1) |
Example values are rounded for readability.
Coordinate rotation in 2D workflows
Point rotation is used in CAD drafting, robotics pose updates, GIS map alignment, and game sprites. This calculator outputs x′ and y′ from x, y, a pivot, and angle, matching the standard 2×2 rotation matrix. When the pivot is the origin, translation terms vanish and the output is a pure linear transform.
Pivot selection and translation impact
For a non‑origin pivot C(cx, cy), the algorithm first translates by Δx = x − cx and Δy = y − cy, then rotates, then adds the pivot back. This three‑step structure keeps results stable for large coordinates because it limits intermediate values to the local offset, not the global position. In practice, offsets reduce overflow risk when coordinates exceed 1e6 units.
Angle units and direction conventions
Angles may be entered in degrees or radians. Internally the calculator uses radians, with θ = degrees × π/180 when degree mode is selected. Counterclockwise rotation uses +θ; clockwise uses −θ. This sign rule matches the right‑handed coordinate convention used in analytic geometry and most plotting libraries. For reference, π radians equals 180°, and 2π equals one full turn.
Precision control and rounding policy
Trigonometric functions return floating values, so outputs are rounded for display using your chosen decimal places. The calculator also normalizes tiny values to 0 to avoid “−0.0000” artifacts. For validation, increase precision to 10–12 decimals and compare against symbolic cases like 90°, 180°, and 360° rotations. Double precision usually keeps relative error near 1e−15 for typical inputs.
Batch processing and export integrity
Batch mode rotates multiple points using identical pivot and angle settings, producing a table for inspection. CSV export includes both the single point and batch rows, suitable for spreadsheets and checks. PDF export captures the same fields plus the formula line, supporting audits and sharing. Use batch lines to spot outliers fast.
Common verification checks and troubleshooting
A quick check is distance preservation: the length from the pivot to the point should match before and after rotation. Another check is quadrant behavior: 90° counterclockwise around the origin maps (a, b) to (−b, a). If results look flipped, confirm direction and unit selection. If scaling appears, verify you did not edit x and y units midstream.
FAQs
Can I rotate around the origin only?
No. Set the pivot to any coordinates. Use (0,0) for origin rotation, or enter cx and cy to rotate around a specific reference point in your diagram.
What is the difference between clockwise and counterclockwise?
Counterclockwise uses a positive angle in the usual coordinate plane. Clockwise flips the sign, so the calculator internally uses −θ while keeping your input angle unchanged.
Why do I see tiny decimals like 0.0000 or −0.0000?
Floating arithmetic can produce very small residuals. Increase precision for diagnostics, or keep lower precision for clean reporting. The tool also snaps extremely small values to zero to remove “−0.0000”.
How do I rotate many points at once?
Paste one point per line in the batch box, using x,y. All points use the same pivot, unit, direction, and precision. The results appear in a table and can be exported to CSV.
How can I verify my result quickly?
Check distance from the pivot: it should be identical before and after rotation. Also test known angles, like 90° mapping (a,b) to (−b,a) around the origin.
Does the calculator support radians directly?
Yes. Choose radians and enter θ in rad units. The calculator will skip conversion and apply trigonometric functions directly, while still supporting clockwise or counterclockwise direction.