2D Point Rotation Calculator

SAME RADIUS / NEW BEARING

Rotate the point without changing its distance from the origin

Use this explainer when you need to see what a 2D rotation preserves, not just the matrix entries. With the reviewed default inputs, the point (3, 4) rotates by 30 degrees to (0.59807621, 4.96410162). The calculator returns x-prime = 0.59807621 as the main result, y-prime = 4.96410162 as the first supporting metric, while the radius stays 5.

Worked default example

Entered point and angle

x = 3, y = 4, theta = 30 degrees.

Returned coordinates

The rotated point is (0.59807621, 4.96410162). Its direction moves from 53.1301 degrees to 83.1301 degrees.

x-prime working

x-prime = x cos(theta) - y sin(theta) = 3 x 0.8660254 - 4 x 0.5 = 0.59807621.

y-prime working

y-prime = x sin(theta) + y cos(theta) = 3 x 0.5 + 4 x 0.8660254 = 4.96410162.

Radius check

Before rotation: sqrt(3^2 + 4^2) = 5. After rotation: sqrt(0.59807621^2 + 4.96410162^2) = 5.

What to enter and how to read the result

Enter the point's x-coordinate, y-coordinate, and a rotation angle in degrees. Positive angles rotate counterclockwise, negative angles rotate clockwise. The main result card shows x-prime, the first metric below it shows y-prime, and together they describe the rotated point about the origin only.

What the calculator actually does

Nirmion applies the standard 2D rotation matrix to the entered Cartesian point: x-prime = x cos(theta) - y sin(theta) and y-prime = x sin(theta) + y cos(theta). In the reviewed default example, cos(30 degrees) = 0.8660254 and sin(30 degrees) = 0.5, so the point direction changes from 53.1301 degrees to 83.1301 degrees while the radius remains 5.

Where this model stops helping

This page rotates one point around the origin in a conventional Cartesian plane. It does not rotate around an arbitrary center, it does not accept radians directly, and it does not model screen-coordinate systems where positive y points downward unless you convert that convention first.

Reference and next checks

The geometry follows the standard rotation formulas summarized in the linked OpenStax section. If you need to validate the radius, convert the angle first, or continue into a broader coordinate problem, use the related calculators below.