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.
Options:
(A) circle
(B) regular octagon
(C) regular pentagon
(D) rhombus
Official Answer Key:
D
Step-by-Step Solution & Explanation
A shape tiles the plane if copies of it can cover the plane with no gaps or overlaps. Regular polygons tile the plane only when their interior angle divides 360° evenly: triangle (60°), square (90°), and hexagon (120°) qualify, but a regular pentagon’s interior angle (108°) and regular octagon’s (135°) do not divide 360° evenly, so they cannot tile alone. A circle always leaves gaps between adjacent circles. A rhombus (a parallelogram), however, always tiles the plane — rhombi can be arranged edge-to-edge in rows to fill the plane completely, exactly like any parallelogram. Answer: (D) rhombus.
Section 1 — Engineering Mathematics
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