GATE 2023 AE – Question 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.
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Correct answer: (D) rhombus
Explanation
A shape tiles the plane without gaps only if the angles meeting at each vertex add up to exactly $360^\circ$.
- **Circle:** circles leave gaps between them. ✗
- **Regular octagon:** each interior angle is $135^\circ$, and $360/135$ is not an integer. ✗ (octagons alone leave square gaps)
- **Regular pentagon:** each interior angle is $108^\circ$, and $360/108$ is not an integer. ✗
- **Rhombus:** a rhombus is a parallelogram, and every parallelogram tiles the plane by repeated translation (its angles are $\theta$ and $180^\circ-\theta$, so two of each meet at a vertex and sum to $360^\circ$). ✓
Answer **rhombus** (D).