Symmetry & Tessellations Cheat Sheet

A printable reference covering line symmetry, rotational symmetry, transformations, regular polygons, angle sums, and tessellation rules for grades 4-8.

Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Symmetry and tessellations help students recognize structure, patterns, and repeated shapes in geometry. This cheat sheet covers line symmetry, rotational symmetry, transformations, and how shapes can tile a plane. Students need these ideas to classify figures, draw accurate patterns, and understand why some shapes fit together without gaps or overlaps. It is especially useful for checking vocabulary, angle rules, and tessellation conditions quickly.

Key Facts

A figure has line symmetry if one line divides it into two matching mirror-image halves.
A figure has rotational symmetry if it matches itself after a turn of less than $360^\circ$ around a center point.
The angle of rotational symmetry for a regular polygon is $\frac{360^\circ}{n}$ , where $n$ is the number of sides.
The sum of the interior angles of a polygon with $n$ sides is $(n - 2) \times 180^\circ$ .
Each interior angle of a regular polygon is $\frac{(n - 2) \times 180^\circ}{n}$ .
Shapes tessellate when the angles meeting at each vertex add to exactly $360^\circ$ with no gaps or overlaps.
All triangles and all quadrilaterals tessellate because copies can be arranged so their angles make $360^\circ$ .
A regular polygon tessellates by itself only when $\frac{360^\circ}{\text{interior angle}}$ is a whole number.

Vocabulary

Line Symmetry: Line symmetry means a figure can be folded along a line so both halves match exactly.
Rotational Symmetry: Rotational symmetry means a figure can be turned around a center point and still look the same before a full $360^\circ$ turn.
Transformation: A transformation is a movement or change of a figure, such as a translation, reflection, rotation, or dilation.
Tessellation: A tessellation is a repeating pattern of shapes that covers a plane with no gaps and no overlaps.
Regular Polygon: A regular polygon is a polygon with all sides congruent and all angles congruent.
Vertex: A vertex is a corner point where two or more sides or edges meet.

Common Mistakes to Avoid

Confusing line symmetry with rotational symmetry is wrong because a mirror fold and a turn are different tests. A shape may have one type of symmetry, both types, or neither.
Counting a full $360^\circ$ turn as rotational symmetry is misleading because every shape matches itself after a full turn. Rotational symmetry must occur before $360^\circ$ .
Assuming all regular polygons tessellate is wrong because the interior angles must fit evenly around a point. For example, a regular pentagon has interior angle $108^\circ$ , and $360^\circ \div 108^\circ$ is not a whole number.
Leaving gaps in a tessellation is wrong because a true tessellation must cover the plane completely. The shapes must meet edge to edge or fit together without empty spaces.
Using side lengths instead of angles to test tessellations is incomplete because angles around each vertex determine whether the shapes fit. The total around a meeting point must be $360^\circ$ .

Practice Questions

1 A regular hexagon has $6$ sides. What is its angle of rotational symmetry using $\frac{360^\circ}{n}$ ?
2 Find the sum of the interior angles of a polygon with $9$ sides using $(n - 2) \times 180^\circ$ .
3 A regular polygon has each interior angle equal to $120^\circ$ . Does it tessellate by itself if $360^\circ \div 120^\circ = 3$ ?
4 Explain why a shape with line symmetry does not always have rotational symmetry.

Sign in to save