Tiling

Class 7 Maths · Chapter 14: Constructions and Tilings · NCERT Ganita Prakash section P2 6.2

Learn

A tangram is a square cut into 7 pieces (5 triangles, a square and a parallelogram). They can be rearranged into animals, people and boats, and every picture has the same area as the square. Tiling means covering a region completely with shapes, with no gaps and no overlaps.

2 × 1 tiles (dominoes): each covers exactly 2 squares of a grid. A 4 × 6 grid (24 squares) can be tiled; a 5 × 7 grid (35 squares, odd) never can. A grid can be tiled exactly when its number of squares is even, which happens when at least one side is even.

5 × 7 = 35 (odd) → impossible

A colouring trick: colour the grid like a chessboard. Each domino covers one black and one white square. Cut two opposite corners off a chessboard: both corners are the same colour, leaving 30 of one colour and 32 of the other. Even though 62 is even, the 31 dominoes can never fit! The same trick shows which square, removed from a 5 × 3 grid, leaves a region that cannot be tiled: if the leftover has 8 of one colour and 6 of the other, it is impossible.

Tiling the whole plane: copies of one shape tile a floor forever only if the angles meeting at each corner add up to 360°. Squares (4 × 90°), equilateral triangles (6 × 60°) and regular hexagons (3 × 120°, like a beehive) work. Regular pentagons (108°) do not: 3 × 108° = 324° leaves a gap. Mixing shapes works too, e.g. two octagons (135° each) and a square (90°). The artist Escher even tiled the plane with animal shapes, and bees build hexagonal cells so that no space is wasted.

135° + 135° + 90° = 360°

Explore: which shapes tile a floor?

1. Sort the shapes: copies of one shape only.

Remember

• Tiling: cover a region with no gaps and no overlaps; 2 × 1 tiles need an even number of squares.
• Shapes tile the plane when the angles meeting at each corner add up to 360°.

Quick check

Can a 3 × 5 grid be covered exactly with 2 × 1 tiles?

1. No: it has 15 squares, an odd number
2. Yes, with 7 tiles
3. Yes, with 8 tiles
4. Only if the tiles are cut

Think about it

Why can't regular pentagons tile a floor by themselves?

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.