Question

Congruent regular pentagons tessellate a plane. true or false?

Answer 1

Congruent regular pentagons cannot tessellate a plane because their interior angles do not fulfill the necessary conditions to fit together without overlapping or leaving gaps. Only regular triangles, squares, and hexagons can tessellate the plane by themselves.

Step-by-step explanation:

The statement that congruent regular pentagons tessellate a plane is false. Tessellation, or tiling, is the covering of a plane using one or more geometric shapes with no overlaps and no gaps. In order for a shape to tessellate on its own, it must be able to fit together with copies of itself to fill the plane.

Regular pentagons do not tessellate because their interior angles, which are each 108 degrees, do not meet the condition of fully filling the space around a point when put together. The angles around a point sum up to 360 degrees. With regular pentagons, five angles sum up to 540 degrees (5 × 108), which is more than 360 degrees, causing an overlap, which breaks the rules of tessellation.

Only regular triangles, squares, and hexagons tessellate the plane by themselves because the interior angles of these shapes are dividers of 360 degrees, making it possible for them to meet at a point without leaving any gaps or causing overlap.

Answer 2

check the tessellation picture below.

now, bear in mind that, you're asked on "regular pentagons", no irregular like the ones in the left-hand-side in the picture. Now, notice the ones in the right-hand-side, those are regular pentagons... well, do they tessellate?