Polygons

Many of the shapes we see around us have mathematical names. Those made of straight lines are called polygons. A polygon is a straight-sided closed geometric figure. The sides may not be curved.

There are an infinite variety of polygons, and as you begin to recognize them, you will see them everywhere: stop signs, snowflakes, roofs, plant forms. The word comes from two Greek words, "poly" meaning "many" and ligonia" meaning "angles". A polygon can have four sides and four angles, or many sides and many angles.

A three-sided polygon is called a triangle, from the Greek word "tri" meaning "three" as in tricycle, or tripod. A three-sided polygon is called a triangle, from the Greek word "tri" meaning "three" as in tricycle, or tripod. It is very easy to find examples of triangles in the world around us, as in this street sign:

A geometric drawing of a triangle is seen below. Every polygon is formed of points and segments joining the points. Each point is called a vertex (the plural for vertex is vertices), and the segments are called the sides of the polygon. The triangle below is called "triangle ABC" using the triangle symbol in the name, as shown.

The triangle is the simplest polygon, and one of the most useful geometric figures. The triangle is also the strongest geometric figure, and is therefore used in the construction of buildings and bridges. Take a drinking straw, and cut it into three pieces. Then thread a piece of string through all three pieces, and tie the ends, tightly, in a knot to create a triangle. Do this again, with a second straw, but cut the second straw into four pieces so that when tied it will form a quadrilateral. Now hold each figure in your hand, and press the top. What happens? The triangle will not change shape, but the quadrilateral will "collapse":

If you tried to build a wooden gate out of a quadrilateral, what would happen to it when you hung the gate on hinges? How would you make it stronger?

By adding another piece of wood, to create two triangles! This is called "triangulating" the structure. Every engineer knows the strength of the triangle.

"Qua" means "four", and a quart is a fourth of a gallon, a quarter is a fourth of a dollar, and a quadrilateral is a four-sided polygon. A square is a special kind of quadrilateral, but a quadrilateral doesn't have to have four right angles nor does it have to have four equal sides. The "Quadrilaterals Tree Diagram below shows you some different types of quadrilaterals and their properties. (Parallel lines are lines that are the same distance apart all along the lines; they don't ever intersect.)

We find quadrilaterals all around us, in everything from fabrics to buildings. The four angles of a rectangle are all 90 degrees and fit together well; lumber stacks into neat piles and rooms fit into rectangular buildings:

Five is an odd number, and a pentagon is a less commonly found polygon than a quadrilateral. If a polygon has all sides equal and all angles equal it is called a regular polygon, even without sides and angles equal it is still a polygon. If the sides are all equal, then the polygon is called just "equilateral". If the angles are all equal, the polygon is called just "equiangular". If a polygon is both equilateral and equiangular, then of course it is regular!

If you drew a polygon that just fit around this Plumeria flower, the polygon would be a pentagon. In Hawaii, we call this flower a Plumeria, and in Tahiti it is called Frangipani.

A polygon with six sides is a hexagon. Six is an easy number to work with, and a hexagon is easy to draw.

We often see hexagons in signs, fabric, even furniture. Every snowflake is a hexagon, and every one is different.

A heptagon is a seven-sided figure, and very rare. Can you find one in your city?

An octagon is much more common, with its eight sides. Designers and artists use octagons in their work. There are many polygons, some regular and many irregular... you see them every day, perhaps without noticing them.

Polygons, and geometry in general, are parts of many different works of art, from architecture to paintings. Op Art is a very interesting type of geometric artwork that produces very unusual optical effects, and images that can "fool the eye". An example of a piece of Op Art, bu an artist named Victor Vasarely, is shown below:

To learn how to construct your own piece of Op Art, click on the link below. The instructions are given for drawing this on a computer, using The Geometer's Sketchpad software, but you can do the same kind of drawing using a pen and a ruler.

OpArtProject.htm


Go to Topic 4 Symmetry