Tessellations & Symmetry Reference Cheat Sheet

A printable reference covering regular and semi-regular tessellations, translations, rotations, reflections, glide reflections, symmetry lines, and rotational symmetry for grades 6-8.

Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

This cheat sheet covers tessellations, symmetry, and the rigid motions used to describe repeating geometric patterns. Students need these ideas to recognize how shapes fill a plane without gaps or overlaps. The reference also helps connect visual patterns to angle measures, transformations, and precise geometry vocabulary. It is designed for quick review during classwork, homework, or test preparation. The most important tessellation rule is that the angles meeting at each vertex must add to $360^\circ$ . For a regular $n$ -gon, the interior angle is $\frac{(n-2)180^\circ}{n}$ , so students can test whether copies of one regular polygon will tessellate. Symmetry describes how a figure can be moved and still match itself, using reflections, rotations, translations, and glide reflections. Rotational symmetry is often measured by the angle $\frac{360^\circ}{n}$ when a figure matches itself $n$ times in one full turn.

Key Facts

A tessellation covers the plane with repeated shapes that have no gaps and no overlaps.
Angles around any point in a tessellation must add to $360^\circ$ .
The interior angle of a regular $n$ -gon is $\frac{(n-2)180^\circ}{n}$ .
A regular polygon tessellates by itself when its interior angle divides evenly into $360^\circ$ .
The only regular polygons that tessellate by themselves are equilateral triangles, squares, and regular hexagons.
A reflection flips a figure over a line so corresponding points are the same perpendicular distance from the line of reflection.
A rotation turns a figure around a fixed center by an angle such as $90^\circ$ , $180^\circ$ , or $270^\circ$ .
If a figure has rotational symmetry of order $n$ , the smallest angle of rotation is $\frac{360^\circ}{n}$ .

Vocabulary

Tessellation: A pattern of shapes that covers a plane completely with no gaps and no overlaps.
Regular tessellation: A tessellation made from only one type of regular polygon, with the same arrangement at every vertex.
Semi-regular tessellation: A tessellation made from two or more regular polygons arranged in the same order at every vertex.
Line symmetry: A figure has line symmetry when a line divides it into two mirror-image halves.
Rotational symmetry: A figure has rotational symmetry when it can be turned less than $360^\circ$ around a center and still match itself.
Glide reflection: A glide reflection is a transformation made by translating a figure and then reflecting it across a line.

Common Mistakes to Avoid

Adding only some of the angles at a vertex is wrong because every angle touching that point must be included to check whether the total is $360^\circ$ .
Assuming every regular polygon tessellates is wrong because the regular polygon's interior angle must divide evenly into $360^\circ$ .
Confusing a translation with a reflection is wrong because a translation slides a figure without flipping it, while a reflection reverses its orientation.
Counting rotational symmetry after a full $360^\circ$ turn only is wrong because every figure matches itself after a full turn, but true rotational symmetry must occur before $360^\circ$ .
Leaving gaps or overlaps in a repeated pattern is wrong because a tessellation must cover the plane completely with no uncovered space and no stacked shapes.

Practice Questions

1 A regular hexagon has interior angle $120^\circ$ . How many regular hexagons meet at one vertex in a regular tessellation?
2 Use $\frac{(n-2)180^\circ}{n}$ to find the interior angle of a regular octagon with $n=8$ .
3 A design has rotational symmetry of order $6$ . What is the smallest angle of rotation that maps the design onto itself?
4 Explain why regular pentagons do not make a regular tessellation, even though they can be repeated in a pattern.

Sign in to save