Math
Polar to Cartesian coordinates
Convert a polar radius and angle into horizontal and vertical coordinates.
Calculators / Math / Rotate a point about the origin
Rotate a point about the origin calculator
Rotate Cartesian coordinates through a signed angle without changing their distance from the origin.
On this page
Rotated x-4
Rotated y3
Distance from origin5
Calculator controls
How to calculate
Rotate Cartesian coordinates through a signed angle without changing their distance from the origin.
Rotated x =
x*cos(a*pi/180)-y*sin(a*pi/180)Rotated y =
x*sin(a*pi/180)+y*cos(a*pi/180)Distance from origin =
sqrt(x^2+y^2)Worked example
- Original x: 3
- Original y: 4
- Rotation angle (degrees): 90
- Rotated x: -4
- Rotated y: 3
- Distance from origin: 5
Assumptions and limits
Positive angles rotate counterclockwise. Rotation is about the origin; shift coordinates first if rotating about another centre.
Results are rounded for display; calculations use unrounded values. Read our calculation methodology.
Understanding the result
Rotate Cartesian coordinates through a signed angle without changing their distance from the origin. A 90-degree positive rotation maps (x, y) to (-y, x). Check that the radial distance is unchanged.
When this tool is useful
A 90-degree positive rotation maps (x, y) to (-y, x). Check that the radial distance is unchanged.
Understanding your inputs
| Input | Example |
|---|---|
Original x x | 3 |
Original y y | 4 |
Rotation angle (degrees) a | 90 |
Frequently asked questions
Which assumptions matter for this result?
Positive angles rotate counterclockwise. Rotation is about the origin; shift coordinates first if rotating about another centre.
How can I check the worked example?
Use the displayed example inputs in the stated formula. Keep each quantity in its labelled unit and compare the unrounded result before applying display rounding.