306

Chapter

Polygons and

Symmetry6

Contents

6-1 Refl ection Symmetry

6-2 Isosceles Triangles

6-3 Angles Inscribed in Circles

6-4 Types of Quadrilaterals

6-5 Properties of Kites

6-6 Properties of Trapezoids

6-7 Rotation Symmetry

6-8 Regular Polygons

6-9 Frieze Patterns

Nuts, bolts and screws are commonly used

to hold things in place. While there are many

different sizes of nuts, there are usually only

two shapes of nuts that can be purchased at

a hardware store, the [regular] hexagonal nut

and the square nut, pictured below. The fi rst

question that comes to mind is, why are those

the only two shapes you can buy?

There are several reasons. First, an adjustable

wrench, like the one pictured above, has

opposite jaws parallel. The jaws grab the nut

and allow a person to turn the nut. That means

that to use an adjustable wrench, the opposite

sides of a nut must be parallel. It is also

reasonable to make the polygon symmetric so

its sides can be grabbed from many angles.

Squares and regular hexagons have opposite

sides that are parallel, while triangles and

pentagons do not.

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307

If parallel sides were the only issue,

then why not make them all squares or

all parallelograms? The answer is that

sometimes it is diffi cult to grab the nut

from a specifi c angle. The hexagon allows

six different angles of approach, while the

square only allows four. In tight places, it is

helpful to have six. But octagons also have

opposite sides parallel and they have eight

angles of approach, so they would seem to

be even better.

The problem with octagons and polygons with

more than six sides is that the measure of

the angle formed by the sides is too large,

so the wrench is likely to slip and might tear

off the corners of the nut. That would be a

huge problem. This leaves only two shapes

workable with a standard wrench.

But there are [regular] pentagonal nuts. You

are likely to fi nd them in fi re hydrants. A fi re

department does not want ordinary citizens

to be able to open fi re hydrants and lower the

water pressure. As a result, they often use

pentagonal nuts to close the hydrant. Then

how do fi remen open the hydrant? They have

a special wrench that is designed to fi t over

the nut, not to its side.

Properties of these and other polygons are

the subject of this chapter.

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