Coordinate Rotation Calculator (2D)

This calculator rotates a point about the origin. Enter its x and y coordinates and the rotation angle in degrees (counterclockwise is positive); you get the coordinates x′ and y′ of the rotated point.

Formula

x′ = x cos α − y sin α and y′ = x sin α + y cos α.

• x, y: the point before rotation
• α: rotation angle in degrees, counterclockwise positive, clockwise negative

Worked Example

Rotate (3, 1) by 30°.

1. cos 30° = 0.866025 and sin 30° = 0.5
2. x′ = 3 × 0.866025 − 1 × 0.5 = 2.0981
3. y′ = 3 × 0.5 + 1 × 0.866025 = 2.3660

How to read the result

The distance from the origin does not change: √(3² + 1²) = √(2.0981² + 2.3660²) ≈ 3.1623. To rotate the axes instead of the point (a change of coordinate system), use −α. To rotate about another centre (h, k), subtract h and k first, rotate, then add them back.

Limits and common mistakes

• Enter degrees, not radians.
• Clockwise turns need a negative angle.
• Values such as cos 90° come out as tiny numbers like 0.0000 rather than exactly 0.

Frequently Asked Questions

What is (1, 0) rotated by 90°?

(0, 1).

How do I rotate clockwise?

Enter a negative angle, for example −45.

Does rotation change the distance from the origin?

No. Rotation keeps lengths and angles.

Is this the same as a rotation matrix?

Yes: [cos α −sin α; sin α cos α] multiplied by the column (x, y).

in x | x | - | 3 | -
in y | y | - | 1 | -
in a | Rotation angle (counterclockwise) | ° | 30 | -
out xr | New x′ | - | x * cosd(a) - y * sind(a) | 4
out yr | New y′ | - | x * sind(a) + y * cosd(a) | 4