About the Rotation, Reflection & Translation Calculator
Transformations move or resize a figure on the coordinate plane. This calculator applies the four you meet in school geometry — translations (slides), rotations (turns), reflections (flips) and dilations (enlargements) — to any list of points, such as the vertices of a triangle or polygon, and returns the image coordinates labelled A′, B′, C′ and so on.
It also writes out the coordinate rule, such as (x, y) → (−y, x) for a 90° counter-clockwise rotation about the origin, and tells you whether the transformation preserves size (an isometry), how the area changes, and whether the orientation of the shape is kept or reversed.
Rotations can be about the origin or any centre point, in either direction and by any angle; reflections cover the x-axis, the y-axis, y = x, y = −x, the origin and any horizontal or vertical line; dilations accept any scale factor and centre. It is aimed at students checking homework and teachers preparing answer keys, as well as anyone placing shapes in design or game coordinates.
How to use the rotation, reflection & translation calculator
- 1List the points (for a shape, its vertices in order), one “x y” pair per line.
- 2Choose rotation, reflection, translation or dilation.
- 3Set the angle and direction, the mirror line, the shift or the scale factor.
- 4For rotations and dilations, set the centre (0, 0 is the origin).
- 5Read the image points A′, B′, C′…, the rule and the properties.
Formula and method
A translation adds the same shift to every point. A rotation first moves the centre to the origin, applies the rotation matrix [cos θ −sin θ; sin θ cos θ] (positive θ is counter-clockwise; clockwise uses −θ), then moves back. For the common angles this gives the familiar rules: 90° counter-clockwise (x, y) → (−y, x), 180° (x, y) → (−x, −y), 270° counter-clockwise (or 90° clockwise) (x, y) → (y, −x).
Reflections swap or negate coordinates: over the x-axis (x, −y), the y-axis (−x, y), y = x (y, x), y = −x (−y, −x), and over x = k or y = k the coordinate becomes 2k minus itself. Translations, rotations and reflections are isometries (they preserve lengths and area); reflections reverse orientation. A dilation multiplies every distance from the centre by the scale factor s, so areas scale by s².
- (x, y)
- Original point (pre-image)
- (x′, y′)
- Transformed point (image)
- a, b
- Horizontal and vertical shift
- θ
- Rotation angle, counter-clockwise positive
- (h, k)
- Centre of rotation or dilation
- s
- Scale factor of a dilation
Worked examples
Rotate triangle (1,2), (4,2), (4,5) by 90° counter-clockwise about the origin
The 90° counter-clockwise rule (x, y) → (−y, x) sends A(1, 2) to (−2, 1), B(4, 2) to (−2, 4) and C(4, 5) to (−5, 4). The triangle keeps its 4.5 square-unit area and its orientation.
Reflect the triangle over the line y = x
Reflecting over y = x swaps the coordinates, so A(1, 2) becomes (2, 1), B(4, 2) becomes (2, 4) and C(4, 5) becomes (5, 4). The image is a mirror image, so its vertex order runs the other way round.
Rotate (5, 3) by 30° clockwise about (2, 1)
Relative to the centre the point is (3, 2). Rotating by −30° gives (3 cos 30° + 2 sin 30°, −3 sin 30° + 2 cos 30°) ≈ (3.598, 0.232), and adding back the centre gives about (5.598, 1.232).
Dilate the triangle by a factor of 2 about the origin
Every coordinate doubles, so the image is (2, 4), (8, 4), (8, 10). Side lengths double and the area grows by 2² = 4, from 4.5 to 18 square units.
Frequently asked questions
What is the rule for a 90° rotation?+
About the origin, a 90° counter-clockwise rotation maps (x, y) → (−y, x), and a 90° clockwise rotation (the same as 270° counter-clockwise) maps (x, y) → (y, −x). A 180° rotation maps (x, y) → (−x, −y) in either direction.
How do you rotate a point around a point other than the origin?+
Subtract the centre from the point, rotate the result about the origin, then add the centre back. For example, rotating (5, 3) by 90° counter-clockwise about (2, 1): (3, 2) → (−2, 3) → (0, 4).
What are the reflection rules over the axes and y = x?+
Over the x-axis (x, y) → (x, −y); over the y-axis (x, y) → (−x, y); over y = x (x, y) → (y, x); over y = −x (x, y) → (−y, −x). Over a vertical line x = k the x-coordinate becomes 2k − x.
Which transformations are rigid motions?+
Translations, rotations and reflections are rigid motions (isometries): they keep every length, angle and area. Dilations keep angles and shape but change size, so they produce similar rather than congruent figures.
What does a negative scale factor do in a dilation?+
It enlarges or shrinks the figure by the absolute value and also sends each point to the opposite side of the centre — the same as a dilation by |s| followed by a 180° rotation about the centre.