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:

For regular polygons to tile a plane by themselves, their interior angle must be a divisor of 360°. The only regular polygons that satisfy this are the equilateral triangle (60°), the square (90°), and the regular hexagon (120°). Any triangle and any quadrilateral (including a rhombus) can also tile a plane.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The concept described is called tessellation or tiling. A shape can tile a plane if identical copies of it can be arranged to fill the plane completely without any gaps or overlaps. A key requirement for this is that the sum of the interior angles of the shapes meeting at any single point (vertex) must be exactly 360 degrees.
Step 2: Detailed Explanation:
Let's analyze each option:
- (A) circle: Circles cannot be placed next to each other without leaving curved, triangular-shaped gaps between them. Therefore, circles cannot tile a plane.
- (B) regular octagon: A regular octagon has 8 equal sides and 8 equal interior angles. The measure of each interior angle is given by the formula \(\frac{(n-2) \times 180^\circ}{n}\), where \(n\) is the number of sides. \[ \text{Angle} = \frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = 135^\circ \] If we try to place regular octagons together at a vertex, the sum of the angles would be \(135^\circ\), \(270^\circ\) (for two), or \(405^\circ\) (for three). None of these sums is exactly \(360^\circ\). Therefore, a regular octagon by itself cannot tile a plane.
- (C) regular pentagon: A regular pentagon has 5 equal sides. The measure of each interior angle is: \[ \text{Angle} = \frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = 108^\circ \] The sum of angles at a vertex would be \(108^\circ\), \(216^\circ\), or \(324^\circ\). Since \(360^\circ\) is not a multiple of \(108^\circ\), a regular pentagon cannot tile a plane.
- (D) rhombus: A rhombus is a quadrilateral (a four-sided polygon) with all four sides of equal length. It is a known geometric fact that any quadrilateral can tile the plane. A rhombus has two pairs of equal opposite angles, say \(\alpha\) and \(\beta\), where \(\alpha + \beta = 180^\circ\). We can arrange copies of the rhombus at a vertex to make the angles sum to \(360^\circ\). For example, we can join several vertices with angle \(\alpha\) until the sum is \(360^\circ\), or do the same for \(\beta\), or a combination. Therefore, a rhombus can always tile a plane.
Step 3: Why This is Correct:
A rhombus, being a quadrilateral, can tessellate the plane. The other shapes listed cannot. Circles leave gaps, and the interior angles of regular octagons and regular pentagons are not divisors of 360°, making it impossible for them to meet at a vertex without gaps or overlaps.