Q.10 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
Answer: (D) rhombus
A rhombus can tile a flat plane completely without gaps or overlaps by repeating identical copies edge-to-edge, as all sides are equal and opposite sides are parallel.
Option Analysis
Circle: Circles leave gaps between them when packed, as their curved edges cannot meet edge-to-edge to fill space perfectly.
Regular Octagon: Regular octagons have 135° interior angles that do not divide evenly into 360° at vertices, creating gaps unless paired with other shapes like squares.
Regular Pentagon: Regular pentagons feature 108° interior angles, which also fail to sum to 360° at meeting points, preventing solo tiling.
Rhombus: Rhombuses fit together seamlessly across the plane, forming parallelogram-like patterns that extend infinitely without voids.
Introduction to Plane Tiling
Tiling a flat plane without gaps involves covering infinite space using identical shapes that fit edge-to-edge, a concept called tessellation. Only certain polygons succeed solo, based on angle sums at vertices reaching exactly 360°. This principle appears in competitive exams testing spatial aptitude.
Why Circles Fail to Tile
Circles cannot tile a flat plane without gaps because their curves create interstitial spaces in any packing arrangement. No edge-to-edge contact occurs, leaving voids everywhere.
Regular Octagon Limitations
A regular octagon’s 135° interior angle means three octagons (405°) exceed 360° at a vertex, while two (270°) leave gaps. Solo tiling fails without squares.
Regular Pentagon Shortcomings
Regular pentagons have 108° angles; three sum to 324° (under 360°), four to 432° (over). This mismatch prevents complete plane coverage alone.
Rhombus Success in Tiling
Rhombuses tile effortlessly: align sides parallel, and flexible angles allow vertex fits totaling 360°. Examples include diamond grids extending infinitely.
