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:

Parallelograms (squares, rectangles, rhombi) always tessellate the plane.
For regular polygons, only equilateral triangles, squares, and hexagons tile by themselves.

Answer 1

The Correct Option is D

Step 1: For a shape to tile the plane, the interior angles must be such that they can fit around a point to sum to \(360^\circ\).
Step 2: A circle cannot tile because gaps remain between circles.
Step 3: A regular octagon cannot tile alone—one needs squares to fill gaps.
Step 4: A regular pentagon cannot tile because interior angle \(108^\circ\) does not divide \(360^\circ\) evenly.
Step 5: A rhombus, being a type of parallelogram, can always tile the plane without gaps. Hence, the answer is (D). \(\boxed{\text{rhombus}}\).