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Constructions and Tilings - Tiling

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Tiling, also known as Tessellation, is the arrangement of shapes to cover a flat surface (plane) without any gaps or overlaps.

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A Regular Tiling is a tiling that uses only one type of regular polygon (where all sides and angles are equal).

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For a regular polygon to tile a plane, the measure of its interior angle must be a factor of 360∘360^\circ. This ensures that when the polygons meet at a vertex, their angles sum up to exactly 360∘360^\circ.

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There are only three regular polygons that can form a regular tiling: Equilateral Triangles (60∘60^\circ), Squares (90∘90^\circ), and Regular Hexagons (120∘120^\circ).

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In any tiling, the point where the corners of the shapes meet is called a vertex.

📐Formulae

Sum of interior angles of a polygon=(n−2)×180∘\text{Sum of interior angles of a polygon} = (n - 2) \times 180^\circ

Interior angle of a regular polygon=(n−2)×180∘n\text{Interior angle of a regular polygon} = \frac{{(n - 2) \times 180^\circ}}{n}

Condition for tiling: 360∘Interior Angle=k (where k is a whole number)\text{Condition for tiling: } \frac{{360^\circ}}{{\text{Interior Angle}}} = k \text{ (where } k \text{ is a whole number)}

💡Examples

Problem 1:

Show why a regular hexagon can tile a floor using its interior angle measure.

Solution:

  1. A regular hexagon has n=6n = 6 sides.
  2. Calculate the interior angle: Interior Angle=(6−2)×180∘6=4×180∘6=120∘\text{Interior Angle} = \frac{{(6 - 2) \times 180^\circ}}{6} = \frac{{4 \times 180^\circ}}{6} = 120^\circ
  3. Check the tiling condition: 360∘120∘=3\frac{{360^\circ}}{120^\circ} = 3 Since 33 is a whole number, 33 hexagons can meet at a vertex to fill the space perfectly.

Explanation:

Because 120∘120^\circ is a divisor of 360∘360^\circ, regular hexagons fit together without gaps.

Problem 2:

Determine if a regular pentagon can form a regular tiling.

Solution:

  1. For a regular pentagon, n=5n = 5.
  2. Calculate the interior angle: Interior Angle=(5−2)×180∘5=3×180∘5=108∘\text{Interior Angle} = \frac{{(5 - 2) \times 180^\circ}}{5} = \frac{{3 \times 180^\circ}}{5} = 108^\circ
  3. Check the tiling condition: 360∘108∘=3.33...\frac{{360^\circ}}{108^\circ} = 3.33... Since 3.33...3.33... is not an integer, 108∘108^\circ is not a factor of 360∘360^\circ.

Explanation:

Regular pentagons cannot tile a surface because they will either overlap or leave a gap at the vertex, as 108∘108^\circ does not divide 360∘360^\circ evenly.

Problem 3:

How many equilateral triangles are required to meet at a single vertex in a regular tiling?

Solution:

  1. For an equilateral triangle, n=3n = 3.
  2. The interior angle is: (3−2)×180∘3=60∘\frac{{(3 - 2) \times 180^\circ}}{3} = 60^\circ
  3. Let kk be the number of triangles: k×60∘=360∘k \times 60^\circ = 360^\circ k=360∘60∘=6k = \frac{{360^\circ}}{60^\circ} = 6

Explanation:

Exactly 66 equilateral triangles meet at each vertex to form a complete tiling.