The most familiar type of symmetry for many people is geometrical symmetry. A geometric object is said to be symmetric if, after it has been geometrically transformed, it retains some property of the original object (i.e., the object has an invariance under the transform). For instance, a circle rotated about its center will have the same shape and size as the original circle. A circle is then said to be symmetric under rotation or to have rotational symmetry.

The type of symmetries that are possible for a geometric object depend on the set of geometric transforms available and what object properties should remain unchanged after a transform. Because the composition of two transforms is also a transform and every transform has an inverse transform that undoes it, the set of transforms under which an object is symmetric form a mathematical group.

The most common group of transforms considered is the Euclidean group of isometries, or distance preserving transformations, in two dimensional (plane geometry)or three dimensional (solid geometry) Euclidean space. These isometries consist of reflections, rotations, translations and combinations of these basic operations.[6] Under an isometric transformation, a geometric object is symmetric if the transformed object is congruent to the original.