Review the key concepts, formulae, and examples before starting your quiz.
🔑Concepts
Tiling (Tessellation) is the process of covering a flat surface using repeating geometric shapes. Imagine a bathroom floor or a street pavement where shapes fit together edge-to-edge; there are no empty spaces and no tiles are placed over each other.
The 'No Gaps, No Overlaps' Rule: For a pattern to be considered a tiling, there must be zero gaps and zero overlaps. If you look at a wall of bricks, they fit perfectly. However, if you try to tile with circles, you will see small diamond-shaped gaps between them because their curved edges do not meet straight.
The Unit Shape: Every tiling is made by repeating a basic 'unit shape' or 'motif.' In a standard grid of squares, the unit shape is a single square. To find the unit shape in a complex design, look for the smallest part of the pattern that repeats over and over.
Shapes that Tile Perfectly: Some shapes are naturally good for tiling. Squares (like on a chessboard), rectangles (like bricks in a wall), and triangles can all cover a surface completely when placed side-by-side.
Hexagonal Tiling (Honeycomb): Regular hexagons are 6-sided shapes that tile perfectly. They create a pattern that looks like a honeycomb made by bees. In this design, three hexagons meet at every corner point, and their sides touch without any space in between.
Non-Tiling Shapes: Not every shape can tile a surface by itself. Circles and ovals are the best examples. Because they are round, they touch at only one point, leaving 'V-shaped' gaps that cannot be filled by other circles of the same size.
Tiling with Multiple Shapes: A design can use more than one type of shape. For example, a floor can be tiled using a pattern of big octagons (8-sided shapes) and small squares to fill the gaps. As long as the entire floor is covered without overlaps, it is a valid tiling.
Color Patterns in Designs: Tilings often use color to create beautiful designs. A simple tiling of squares becomes a checkerboard pattern when we use two colors, like black and white, and alternate them so that no two tiles of the same color touch along an edge.
📐Formulae
Total Area of Tiling =
Area of a square tile =
Number of tiles needed =
Sum of angles at a point in a tiling =
💡Examples
Problem 1:
Riya wants to cover a rectangular table top that is long and wide. If she uses square tiles of size unit, how many tiles does she need?
Solution:
Step 1: Find the total area of the table top. square units. Step 2: Since each square tile has an area of square unit, divide the total area by the tile area: .
Explanation:
Because squares tile perfectly without gaps or overlaps, the number of tiles is simply the total area of the surface divided by the area of a single tile.
Problem 2:
In a tiling made of squares, squares meet at every corner. If each corner angle of a square is , prove that there are no gaps at the meeting point.
Solution:
Step 1: Identify the number of angles meeting at the point, which is . Step 2: Multiply the number of angles by the measurement of each angle: .
Explanation:
For shapes to fit perfectly around a point without any gaps, the sum of the angles meeting at that point must be exactly . Since squares provide exactly , they tile perfectly.