Reflection Calculator

A reflection is a transformation that creates a mirror image of a point or shape across a line called the line of reflection. Every point in the original figure is mapped to a corresponding point on the opposite side of the line, at an equal perpendicular distance. The result is a congruent image that appears 'flipped'. See also our Shoelace Formula Calculator.

To reflect a point P(x, y) across the x-axis, keep the x-coordinate the same and negate the y-coordinate. The reflected point is P′(x, −y). For example, reflecting (3, 4) across the x-axis gives (3, −4).

Reflecting across the y-axis keeps the y-coordinate unchanged and negates the x-coordinate. The rule is P(x, y) → P′(−x, y). For example, (3, 4) becomes (−3, 4) after reflection over the y-axis.

Reflecting through the origin negates both coordinates: P(x, y) → P′(−x, −y). This is equivalent to a 180° rotation about the origin. For example, (3, 4) reflected through the origin gives (−3, −4). You might also find our Triangulation useful.

When reflecting across y = x, the x and y coordinates are simply swapped: P(x, y) → P′(y, x). So (3, 4) becomes (4, 3). This line acts as a diagonal mirror through the origin at a 45° angle.

Reflecting across y = −x swaps the coordinates and negates both: P(x, y) → P′(−y, −x). For example, (3, 4) reflected across y = −x gives (−4, −3). This line runs diagonally through the origin in the opposite direction from y = x.

The distance between an original point and its reflected image equals twice the perpendicular distance from the original point to the line of reflection. The midpoint of the segment connecting P and P′ always lies exactly on the line of reflection.

This calculator reflects individual points. To reflect a triangle or polygon, apply the same reflection rule to each vertex separately, then connect the reflected vertices in the same order as the original shape. The reflected shape will be congruent to the original.