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 not allowed to overlap.

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

Hint:

All parallelograms (including rectangles, squares, and rhombuses) tessellate the plane. Regular polygons tessellate only if their angles divide \(360^\circ\) evenly (like triangles, squares, hexagons).

Answer 1

The Correct Option is D

Step 1: Recall tiling (tessellation) rules.
A shape can tessellate (tile) the plane if copies can cover the entire plane without gaps or overlaps. Typically, only shapes whose angles are divisors of \(360^\circ\) can tile regularly.

Step 2: Analyze each option.
(A) Circle — Circles leave gaps when placed side by side; they cannot perfectly tessellate the plane. ✗ (B) Regular octagon — Regular octagons alone cannot tile the plane. They leave gaps, which require squares to fill. Hence, they do not tessellate by themselves. ✗ (C) Regular pentagon — Regular pentagons cannot tile the plane because their interior angle is \(108^\circ\), which does not divide \(360^\circ\). ✗ (D) Rhombus — A rhombus (a type of parallelogram) tiles the plane perfectly because parallelograms tessellate the plane without gaps or overlaps. ✓

Step 3: Conclusion.
The only option that satisfies the condition is the rhombus.

Final Answer:
\[ \boxed{\text{Rhombus (Option D)}} \]