Point Rotation Calculator

How it is calculated

The point (x, y) is rotated about the origin (0, 0) by an angle θ in degrees. A positive angle turns the point counter-clockwise and a negative angle turns it clockwise. The tool converts the angle to radians and applies x′ = x cos θ − y sin θ and y′ = x sin θ + y cos θ. Both new coordinates are rounded to two decimals. The distance from the origin, √(x² + y²), stays the same, which is a good check on the result. Any real x, y and angle is accepted, and angles beyond 360 simply wrap around, so 405 gives the same point as 45. To rotate about another centre, subtract that centre from the point first and add it back afterwards.

Common mistakes

• Entering the angle in radians. The field is degrees, so 1.57 for a quarter turn gives a rotation of under 2 degrees.
• Expecting a positive angle to turn clockwise. Positive is counter-clockwise, as in the standard x-y plane, so use a negative angle for clockwise.
• Rotating about a point other than the origin without adjusting. The tool always uses (0, 0), so shift the point by the pivot first and shift back afterwards.
• Screen coordinates with y pointing down. On a screen the same angle looks clockwise, so the direction can seem reversed.
• Confusing the old and new coordinates. The two outputs are the rotated x and y, and the original point is not changed.