The table shows the interior angles for different regular polygons. Number of sides Interior angle 3 60 ° 4 90 ° 5 108 ° 6 120 ° 7 128 . 6 ° 8 135 ° Which are the only regular polygons to tessellate?

The table shows the interior angles for different regular polygons. Number of sides Interior angle 3 60 ° 4 90 ° 5 108 ° 6 120 ° 7 128 . 6 ° 8 135 ° We know that a regular polygon can only tessellate if its interior angle is a factor of 360 ° . Here, 108 ° , 128 . 6 ° and 135 ° are not factors of 360 ° . Therefore, the regular polygons with sides 5 , 7 and 8 will not tessellate because their interior angles are not factors of 360 ° . Here, 60 ° , 90 ° and 120 ° are factors of 360 ° . Therefore, the regular polygons with sides 3 , 4 and 6 will tessellate because their interior angles are factors of 360 ° .

Simone tries to tessellate this rhombus. She puts three tiles together as shown below: Can she continue her pattern to make a tessellation of her rhombus? Explain your answer.

Simone tries to tessellate this rhombus. She puts three tiles together as shown below: Let us name the three tiles put together by Simone as shown below. Looking at the figure, we can see that there is a gap between tile A and tile B . We know that a shape can be tessellated if it can be used to make a repeating pattern with no overlaps or gaps. Therefore, Simone cannot continue with the pattern because she cannot make a repeating pattern with no overlaps or gaps.

Hazeem tries to tessellate this isosceles triangle. He puts six tiles together as shown. Show that Hazeem won't be able to continue his pattern to make a tessellation.

Name the topmost triangle as shown below: Rotate the triangle by 180 ° about the midpoint of A C as shown. Again, rotate the triangle by 180 ° about the midpoint of A B as shown below. In the above figure, we see that the triangles overlap each other. We know that a shape can be tessellated if it can be used to make a repeating pattern with no overlaps or gaps. Therefore, Hazeem won't be able to continue his pattern to make a tessellation.