Area and Tilings
A Lesson in the “Math + Fun!” Series
June 2007
Area and Tilings
Slide 1
About This Presentation
This presentation is part of the “Math + Fun!” series devised
by Behrooz Parhami, Professor of Computer Engineering at
University of California, Santa Barbara. It was first prepared
for special lessons in mathematics at Goleta Family School
during four school years (2003-07). “Math + Fun!” material
can be used freely in teaching and other educational settings.
Unauthorized uses are strictly prohibited. © Behrooz Parhami
June 2007
Edition
Released
First
June 2007
Revised
Area and Tilings
Revised
Slide 2
Finding the Area of a Geometric Shape
Square
Rectangle
Circle
Area = Side2
Area = Width  Height
Area = (p/4)  Diameter2
= p  Radius2
Triangle
Area = Base  Height / 2
Triangle
Area = Base  Height / 2
The area of a circle is about 80% of the square that encloses it
The area of a triangle is half that of a rectangle that encloses it
June 2007
Area and Tilings
Slide 3
Activity 1: The Area of a Triangle
On a sturdy piece of cardboard, draw a 4”  6” rectangle
Place thumbtacks or pushpins into the two lower corners
Put a rubber band around the two thumbtacks or pushpins
and stretch it so that it forms a triangle, with the top vertex
at the upper left corner of the rectangle. Is it obvious that
the area of the triangle is half the area of the rectangle?
Now, slowly move the rubber band so that the top vertex shifts
to the right along the rectangle’s top edge. What happens to
the area of the triangle as you move the top vertex?
June 2007
Area and Tilings
Slide 4
Measuring the Area of an Irregular Shape
Method 1: Approximate the irregular shape by a regular one
Method 2: Cover with 1  1 tiles; count whole tiles and half of broken ones
June 2007
Area and Tilings
Slide 5
Activity 2: Tiling an Area with Square Tiles
Draw a large irregular area on a piece
of cardboard or construction paper
Draw a straight line through the
middle of the area in any direction
Use square
post-it notes
as your tiles
15” or more
Place tiles, one by one, on one side of the straight line that you have drawn,
taking care that the tiles are aligned and there is no gap between them
(real tilers actually leave a gap between tiles where they will pour the grout)
Now, moving up and down from the row of tiles placed next to the line,
finish tiling of the area, leaving spaces only where whole tiles would not fit;
make sure the tile sides are perfectly aligned, with no gap between them
Cut tiles to appropriate shapes to fill the irregular areas at the edges
Taking your tiles to be 1’  1’, estimate the area of the irregular shape in ft2
June 2007
Area and Tilings
Slide 6
Simple Tilings with Nonsquare Tiles
Any shape with right angles and side lengths that
are integers can be tiled using 1  1 tiles.
Some, but not all, shapes can be tiled using 1  2 tiles
To be completely covered with 1  2 tiles, a shape’s area must be even,
but this is not enough
June 2007
Area and Tilings
Slide 7
Covering a Chess Board with 1  2 Tiles
A chess board, or any rectangle with at least one even side,
can be completely covered with 1  2 tiles
What if we remove two squares at opposite corners?
June 2007
Area and Tilings
Slide 8
Activity 3: Tiling with 1  2 Tiles
Tile a 4  6 rectangle using 1  2 tiles of two
different colors. Try to find at least two tilings
that look nice (have interesting color patterns)
June 2007
Area and Tilings
Slide 9
Activity 4: Tiling with L-Shaped Tiles
Tile a 4  6 rectangle using L-shaped
tiles that cover three squares.
Is there more than one way to do this?
June 2007
Area and Tilings
Slide 10
Some Possible 1  2 Tiling Patterns
Mixed with 1 x 1
Challenge: Try to come up with other ways of mixing 1  2 and 1  1 tiles
June 2007
Area and Tilings
Slide 11
Some Irregular Tiling Patterns
Challenge: Try to come up with other interesting irregular tiling patterns
June 2007
Area and Tilings
Slide 12
Triangular, Hexagonal, and Other Patterns
These mixed hexagonal
and pentagonal tiles
don’t quite cover a flat
surface area but . . .
June 2007
Area and Tilings
Slide 13
Activity 5: Mixed Triangular and Hexagonal Tiles
Cut out a number of hexagonal and triangular tiles with sides of equal
length (use paper of different colors) and use them to tile a square area
June 2007
Area and Tilings
Slide 14
Two-Color Tiles
June 2007
Area and Tilings
Slide 15
Multicolor and Patterned Tiles
June 2007
Area and Tilings
Slide 16