Question

Which one of the following shapes can be used to tile (completely cover by repeating) a flat plane, extending to infinity in all directions, without leaving any empty spaces in between them? The copies of the shape used to tile are identical and are not allowed to overlap.

• A. circle
• B. regular octagon
• C. regular pentagon
• D. rhombus

Hint:

For regular $n$-gons to tile alone, the interior angle $\frac{(n-2)180^\circ}{n}$ must divide $360^\circ$. Only $n=3,4,6$ work. Parallelograms (including rhombi) always tile by translation.

Answer 1

The Correct Option is D

A shape tiles the plane if copies meet edge-to-edge with interior angles around every vertex summing to $360^\circ$.
- Circle: cannot tile because curved boundaries leave gaps unless stretched/overlapped. No.
- Regular octagon: interior angle $=135^\circ$; $360/135=2.666\ldots$ is not an integer, so identical regular octagons alone cannot fill the plane (they need squares). No.
- Regular pentagon: interior angle $=108^\circ$; $360/108=3.\overline{3}$ not an integer; identical regular pentagons cannot tile the plane. No.
- Rhombus (any parallelogram with equal sides): opposite angles are equal and adjacent angles sum to $180^\circ$; two copies meet to make $360^\circ$ around each vertex via translations. All parallelograms, hence rhombi, tile the plane. Yes.
\[ \boxed{\text{Rhombus tiles the plane}} \]