Rotating Around a Point Calculator

Rotate coordinates around any pivot using clear step outputs. Check direction and transformed values instantly. Plot results, export tables, and verify geometry with confidence.

Calculator Form

Point X

Point Y

Center H

Center K

Angle

Decimal Precision

Angle Unit

Direction

Reset

Example Data Table

Point	Center	Angle	Direction	Rotated Point
(4, 2)	(1, 1)	90 degrees	Counterclockwise	(0, 4)
(5, 1)	(2, 1)	180 degrees	Counterclockwise	(-1, 1)
(3, 4)	(0, 0)	45 degrees	Clockwise	(4.95, 0.71)
(2, 5)	(2, 2)	270 degrees	Counterclockwise	(5, 2)

Formula Used

Translate the point so the rotation center becomes the temporary origin.

dx = x - h

dy = y - k

Use a signed angle. Counterclockwise stays positive. Clockwise becomes negative.

x′ = h + dx cos(θ) - dy sin(θ)

y′ = k + dx sin(θ) + dy cos(θ)

This also matches the rotation matrix below.

[x_r] = [cos(θ) -sin(θ)] [dx]

[y_r] [sin(θ) cos(θ)] [dy]

Then shift back by adding the center coordinates.

How to Use This Calculator

Enter the original point coordinates.
Enter the rotation center coordinates.
Provide the angle value.
Choose degrees or radians.
Select clockwise or counterclockwise rotation.
Set the decimal precision you need.
Press Calculate to view the rotated point above the form.
Use the graph and export buttons for checking and reporting.

FAQs

1. What does rotating around a point mean?

It means moving a point around a chosen center by a fixed angle while keeping the same distance from that center. The point changes direction, not radius.

2. What is the difference between clockwise and counterclockwise?

Counterclockwise uses a positive angle in standard coordinate geometry. Clockwise uses the same magnitude but a negative signed angle in the formula.

3. Can I use degrees and radians?

Yes. Pick the correct unit before calculating. The calculator converts degrees internally when needed and uses the same rotation formula afterward.

4. Why does the distance from the center stay unchanged?

Rotation is a rigid transformation. It preserves length, so the point keeps the same radius from the selected center before and after rotation.

5. Can I rotate around the origin with this tool?

Yes. Enter 0 for both center coordinates. The formula becomes the standard origin rotation form used in coordinate geometry problems.

6. Why do I get decimal results?

Most rotations use sine and cosine values that are irrational or non-integer. Decimal output gives a practical approximation based on your chosen precision.

7. What does decimal precision change?

Precision controls how many digits appear after the decimal point in outputs. It helps when you need cleaner classroom values or more exact engineering-style results.

8. How does the graph help?

The graph shows the center, original point, rotated point, and path guide. It helps confirm direction, angle sense, and the preserved distance visually.

Notes

This calculator is useful for geometry exercises, coordinate transformations, computer graphics preparation, and checking manual rotation steps.