Table of Contents

## What polygon will tessellate by itself?

Equilateral triangles, squares and regular hexagons are the only regular polygons that will tessellate.

**What shape will not tessellate by itself?**

Circles or ovals, for example, cannot tessellate. Not only do they not have angles, but you can clearly see that it is impossible to put a series of circles next to each other without a gap. See? Circles cannot tessellate.

**What polygons Cannot tessellate?**

Only three regular polygons tessellate: equilateral triangles, squares, and regular hexagons. No other regular polygon can tessellate because of the angles of the corners of the polygons. This is not an integer, so tessellation is impossible. Hexagons have 6 sides, so you can fit hexagons.

### Will a regular octagon tessellate alone?

No, a regular octagon cannot tessellate.

**Can you tessellate six sided regular polygons by themselves?**

Since the interior angles get larger as the number of sides in a polygon gets larger, no regular polygons with more than six sides can tessellate by themselves. Since there are no integers between three and four, pentagons must not tessellate.

**Can a regular pentagon tessellate?**

Regular tessellation We have already seen that the regular pentagon does not tessellate. A regular polygon with more than six sides has a corner angle larger than 120° (which is 360°/3) and smaller than 180° (which is 360°/2) so it cannot evenly divide 360°.

#### How can you tell if a shape will tessellate?

Regular polygons tessellate if the interior angles can be added together to make 360°.

- A square has an interior angle of 90°, so 4 squares fit together to make 360°: 360 ÷ 90 = 4.
- An equilateral triangle has an interior angle of 60°, so 6 triangles fit together to make 360°: 360 ÷ 60 = 6.

**Can hexagons tessellate?**

Triangles, squares and hexagons are the only regular shapes which tessellate by themselves. You can have other tessellations of regular shapes if you use more than one type of shape.

**Can a regular Pentagon tessellate?**

## Why do hexagons tessellate but not pentagons?

there is a regular tessellation using three hexagons around each vertex. We have already seen that the regular pentagon does not tessellate. A regular polygon with more than six sides has a corner angle larger than 120° (which is 360°/3) and smaller than 180° (which is 360°/2) so it cannot evenly divide 360°.

**Can a Hendecagon tessellate?**

A regular decagon does not tessellate. A regular polygon is a two-dimensional shape with straight sides that all have equal length.

**Can You tessellate a pentagon with three hexagons?**

there is a regular tessellation using three hexagons around each vertex. We have already seen that the regular pentagon does not tessellate. A regular polygon with more than six sides has a corner angle larger than 120° (which is 360°/3) and smaller than 180° (which is 360°/2) so it cannot evenly divide 360°.

### Are there any polygons that do not tessellate the plane?

In fact, there are pentagons which do not tessellate the plane. For example, the regular pentagon has five equal angles summing to 540°, so each angle of the regular pentagon is \\frac {540^\\circ} {5} = 108^\\circ . Attempting to fit regular polygons together leads to one of the two pictures below:

**How do you make a tessellation of a quadrilateral?**

All quadrilaterals tessellate. Begin with an arbitrary quadrilateral ABCD. Rotate by 180° about the midpoint of one of its sides, and then repeat using the midpoints of other sides to build up a tessellation. The angles around each vertex are exactly the four angles of the original quadrilateral.

**Which is the sum of angles of a polygon tessellation?**

The resulting parallelogram tessellates: The picture works because all three corners (A, B, and C) of the triangle come together to make a 180° angle – a straight line. This property of triangles will be the foundation of our study of polygon tessellations, so we state it here: The sum of angles of any triangle is 180°.