Using Books to Draw Shapes

In classrooms around the world the blackboard remains the major form of visual communication. In many of these classrooms teachers have few if any geometrical instruments. Here are some ideas on drawing simple mathematical shapes and constructions using only what is available everywhere…. the book.

As this picture shows, drawing around a book is an ideal

way to create a simple rectangle or right angle.

Drawing around the book several times can help us in the drawing of clear accurate fraction pictures.

Drawing shapes

You might like to try this as a challenge first. Can you use books to construct

· An equilateral triangle?

· A square?

· A kite?

· A rhombus?

· An octagon?

· A pentagon?

· A hexagon?

Look below for some ideas.

Using three similar books an equilateral triangle can be constructed, though you may need someone to help hold two of the books!

Constructing a Square

Step 1 Draw around three sides of the book.

Step 2 Then rotate the book and place it in a corner already drawn.

Draw around the sides to complete another corner.

Rotate once more and draw along the edges to complete the square.

Drawing a Rhombus

Draw along two opposite sides of the book. Now twist the book and draw along the sides again.

This creates two more parallel lines that cross the first pair. Finally rub out the lines that are not needed and you are left with a rhombus.

Drawing a Kite

Draw along two adjacent edges of the book. Rotate the book and place opposite corners to lie at the ends of the lies already drawn.

Draw along two sides to create a kite.

Drawing an Octagon

Here is a method that uses the fact that many books are A4 in size.

A) First draw around four sides of the book

B) Next rotate the book 45 degrees so that the corners fit.

Draw along sides of the book to give three more sides of the octagon .

C) Rotate the book 90 degrees so it fits in the first outline.

Draw two more sides of the octagon .

D) Finally turn the book once more through 45 degrees to give the final side.

Rub out the unwanted lines to reveal the octagon.

If you are not sure if the book you are using is “A” size then use this test to find out.

Here is an alternative way of constructing an Octagon

To complete the octagon continue on from Step 4 five more times.

If you use the same method but this time draw around all sides of the book in each position then a dramatic octagon with internal stars is created.

Constructing a Pentagon

This method once again makes use of the A4 rectangle and gives a nearly perfect pentagon in a very short time.

Step 1 First draw along the base of the book and mark the position of the top right corner. This point A.

Step 2 Then rotate the book so that opposite corners lie as shown.

One is at the left end of the base and the other on the point A previously marked.

Draw along the base of the book and once again mark the position of the top right corner of the book (point B).

Step 3 Repeat this procedure using the side just completed and point B to help position the book. Now draw along the base of the book to create a new side of the pentagon and mark point C.

Step 4 Repeat this same procedure using point C to help place the book and draw another side. Then complete the pentagon as shown in Step 5.

It may not be quite perfect but is close to a regular pentagon.

Constructing a Hexagon

There seems to me no really simple way of creating a hexagon using books that is practical to use on the board.

Here, however, is a method that makes use of the fact that if the books are A4 in size then the ratio of sides is 1:/2 and the ratio of the short side to the diagonal is 1:/3.

Step 1 First mark points at opposite corners of the book.

Step 2 Next using two books create an equilateral triangle with this line as base.

Step 3 Now use the special property of A size and the fact that the ratio of the sides of a hexagon to its height is 1 : /3. Use pairs of books with their short sides placed between the corners of the equilateral triangle to draw the sides of the hexagon.

Step 4 Repeat step 3 to create the remaining two sides

It is also possible to use books to do those geometrical constructions we usually do with the help of a pair of compasses.

Can you use books to

· Bisect an acute angle?

· Bisect an obtuse angle?

· Bisect a line?

Below are some ideas.

Constructions with a book

Bisecting an acute angle

Place the book along one line that makes the angle so that the corner of the book lies on the vertex.

Then draw along the other edge to create a perpendicular.

Repeat this on the other arm of the angle so that you have created two perpendiculars which are an equal distance from the vertex.

Then join the point of intersection of the two perpendiculars to the vertex and this line bisects the angle.

Bisecting an obtuse angle.

Use the book to draw two lines parallel to the arms of the angle.

Then join the vertex of the angle to the point where these lines intersect.

Bisecting a line

Place the book on the line so that opposite corners lie on the line and one corner is at the end of the line.

Mark the edges of the book and repeat above and below at both ends of the line.

Join the intersections to create the perpendicular bisector.

If the line is too short or too long for this method to work then variations can be designed which will work. For example, place the book at each end of the line and draw along the side. Then join the ends of the two perpendiculars using a straight edge or ruler.

These ideas first appeared in Mathematics in School, March 1999.