**Introduction to Polygons**

There is no limit to the number of sides a polygon can have. In order to qualify as a plane polygon, all of the vertices (points where the sides come together) must lie in the same plane, and no two sides are allowed to cross over each other. No two vertices can coincide. No three vertices can lie on a common line (otherwise we might get confused as to whether a line segment represents one side or two). And finally, the sides must all be straight line segments having finite length. They can’t be curved, and they can’t go off into infinity.

As you can guess, plane polygons get increasingly complicated as the number of sides increases. Let’s consider a few special cases.

**The Regular Pentagon**

Figure 4-1 shows a five-sided polygon, all of whose sides have the same length, and all of whose interior angles have the same measure. This is called a *regular pentagon* . It is called *convex* because its exterior never bends inward. Another way of saying this is that all of the interior angles measure less than 180° ( *π* rad).

**Fig. 4-1** . A regular pentagon. Each side is *s* units long, and each interior angle has measure *z* .

**The Regular Hexagon**

A convex polygon with six sides, all of which are equally long, is called a *regular hexagon* (Fig. 4-2). This type of polygon is common in nature. If there are many of them and they are all the same size, they can be placed neatly together without any gaps. (Do you remember those old barbershops where the floors were made of little hexagonal tiles that fit up against each other?) This makes the regular hexagon a special sort of figure, along with the equilateral triangle and the square. Certain crystalline solids form regular hexagonal shapes when they fracture. Snowflakes have components with this shape.

**Fig. 4-2** . A regular hexagon. Each side is *s* units long, and each interior angle has measure *z* . The extensions of sides (dashed lines) are the subject of Problem 1.

**The Regular Octagon**

Figure 4-3 shows a *regular octagon* . This is a convex polygon with eight sides, all equally long. As is the case with the regular hexagon, large numbers of these figures can be fit neatly together. So it is not surprising that nature has seen fit to take advantage of this, building octagonal crystals.

**Fig. 4-3** . A regular octagon. Each side is *s* units long, and each interior angle has measure *z* .

**Regular Polygons In General**

For every whole number *n* greater than or equal to 3, it is possible to have a regular polygon with *n* sides. So far we’ve seen the equilateral triangle ( *n* = 3), the square ( *n* = 4), the regular pentagon ( *n* = 5), the regular hexagon ( *n* = 6), and the regular octagon ( *n* = 8). There can exist a regular polygon with 1000 sides (this might be called a “regular kilogon”), 1,000,000 sides (a “regular megagon”), or 1,000,000,000 sides (a “regular gigagon”). These last three would look pretty much like circles to the casual observer.

**General, Many-sided Polygons**

Once the restrictions are removed concerning the relationship among the sides of a polygon having four sides or more, the potential for variety increases without limit. Sides can have all different lengths, and the measure of each interior angle can range anywhere from 0° (0 rad) to 360° (2 *π* rad), non-inclusive.

Figure 4-4 shows some examples of general, many-sided polygons. The object at the top left is a non-convex octagon whose sides happen to all have the same length. The interior angles, however, differ in measure. The other two objects are irregular and non-convex. All three share the essential characteristics of a plane polygon:

**Fig. 4-4** . General, many-sided polygons. The object with the shaded interior is the subject of Problem 2.

- The vertices all lie in a single plane
- No two sides cross
- No two vertices coincide
- No three vertices lie on a single straight line
- All the sides are line segments of finite length

**Polygons, Five Sides and Up Practice Problems**

**PROBLEM 1**

What is the measure of each interior angle of a regular hexagon?

**SOLUTION 1**

Draw a horizontal line segment to start. All the other sides must be duplicates of this one, but rotated with respect to the first line segment by whole-number multiples of a certain angle. This rotation angle from side to side is 360° divided by 6 (a full rotation divided by the number of sides), or 60°. Imagine the lines on which two adjacent sides lie. Look back at Fig. 4-2. These lines subtend a 60° angle with respect to each other, if you look at the acute angle. But if you look at the obtuse angle, it is 120°. This obtuse angle is an interior angle of the hexagon. Therefore, each interior angle of a regular hexagon measures 120°.

**Fig. 4-2** . A regular hexagon. Each side is *s* units long, and each interior angle has measure *z* . The extensions of sides (dashed lines) are the subject of Problem 1.

**PROBLEM 2**

Briefly glance at the lowermost polygon in Fig. 4-4 (the one with the shaded interior). Don’t look at it for more than two seconds. How many sides do you suppose this object has?

**Fig. 4-4** . General, many-sided polygons. The object with the shaded interior is the subject of Problem 2.

**SOLUTION 2**

This is an optical illusion. Most people underestimate the number of sides in figures like this. After you’ve made your guess, count them and see for yourself!

Practice problems for these concepts can be found at: Other Plane Figures Practice Test.