A rotation calculator moves a point in the plane around the origin by a chosen angle. It is used in geometry homework, computer graphics, and robotics, where a shape must be turned without changing its size or shape. The output is a new pair of coordinates, not a drawing, so it is easy to check and easy to reuse in a larger problem.
Rotation preserves distance from the origin and preserves the shape of a figure. That is why the same formula works for a single point, for every vertex of a polygon, and for a sprite on a screen. If you rotate each vertex of a triangle by the same angle around the same centre, the triangle turns as a whole and its side lengths stay the same.
What this calculator does
For a point (x, y) rotated counterclockwise by angle t around the origin, the new coordinates are x’ = x cos t – y sin t and y’ = x sin t + y cos t. The cosine and sine must use the same angle, and the angle must be in the correct unit. This calculator accepts degrees and converts to radians internally, because most geometry problems state angles in degrees while most programming libraries expect radians.
The minus sign in the x’ formula is easy to miss, and it is what makes rotation different from a simple scaling. For a 90-degree counterclockwise turn, cos t is 0 and sin t is 1, so the formulas collapse to x’ = -y and y’ = x. That special case is a quick check: a point on the positive x-axis should move to the positive y-axis after a 90-degree counterclockwise rotation.
Formula
Counterclockwise by angle t: x’ = x cos t – y sin t, y’ = x sin t + y cos t. Clockwise by t is the same as counterclockwise by -t.
How to use it
- Enter the original x and y coordinates.
- Enter the angle in degrees, positive for counterclockwise and negative for clockwise.
- Convert degrees to radians only if you are calculating by hand; the calculator does that step for you.
- Apply the two formulas, keeping the minus sign in the x’ formula.
- Check the distance from the origin before and after. It must stay the same.
Try the calculator
Worked example
Rotate the point (3, 4) by 90 degrees counterclockwise around the origin. Using cos 90 = 0 and sin 90 = 1, x’ = 3 x 0 – 4 x 1 = -4, and y’ = 3 x 1 + 4 x 0 = 3. The rotated point is (-4, 3). As a check, the distance from the origin before rotation is sqrt(3^2 + 4^2) = 5, and after rotation it is sqrt((-4)^2 + 3^2) = 5. The distance is unchanged, which is exactly what a rotation should do.
A 180-degree rotation gives an even simpler check: (x, y) becomes (-x, -y). For (3, 4), that is (-3, -4), again at distance 5 from the origin.
How to check your answer
Calculate the distance from the origin before and after rotation. A changed distance means an arithmetic error, not a rotation.
Test a 90-degree or 180-degree case by hand, where sine and cosine are 0, 1, or -1 and the arithmetic is simple.
Rotate back by the negative angle. You should return to the original point.
Common mistakes
- Using degrees in a formula or calculator mode set to radians, or the reverse.
- Dropping the minus sign in x’ = x cos t – y sin t.
- Rotating clockwise when the problem asks for counterclockwise, which gives the mirror result across the line y = x in some cases.
- Assuming rotation is around the point itself rather than the origin, unless the problem states another centre.
Frequently asked questions
What does a negative angle mean?
A negative angle rotates clockwise. Rotating by -90 degrees is the same as rotating clockwise by 90 degrees, and the formulas handle the sign automatically.
Does rotation change the size of a shape?
No. Rotation is a rigid motion, so distances and angles within the shape stay the same. Only position and orientation change.
How do I rotate around a point other than the origin?
Translate the centre to the origin, rotate, then translate back. For centre (h, k), rotate (x – h, y – k) and then add (h, k) to the result.
