# Geometry

## Polygons, Perimeter, and Circumference


### Learning Objectives

After completing this section, you should be able to:
1. Identify polygons by their sides.
2. Identify polygons by their characteristics.
3. Calculate the perimeter of a polygon.
4. Calculate the sum of the measures of a polygon’s interior angles.
5. Calculate the sum of the measures of a polygon’s exterior angles.
6. Calculate the circumference of a circle.
7. Solve application problems involving perimeter and circumference.
In our homes, on the road, everywhere we go, polygonal shapes are so common that we cannot count the many uses. Traffic signs, furniture, lighting, clocks, books, computers, phones, and so on, the list is endless. Many applications of polygonal shapes are for practical use, because the shapes chosen are the best for the purpose.
Modern geometric patterns in fabric design have become more popular with time, and they are used for the beauty they lend to the material, the window coverings, the dresses, or the upholstery. This art is not done for any practical reason, but only for the interest these shapes can create, for the pure aesthetics of design.
When designing fabrics, one has to consider the perimeter of the shapes, the triangles, the hexagons, and all polygons used in the pattern, including the circumference of any circular shapes. Additionally, it is the relationship of one object to another and experimenting with different shapes, changing perimeters, or changing angle measurements that we find the best overall design for the intended use of the fabric. In this section, we will explore these properties of polygons, the perimeter, the calculation of interior and exterior angles of polygons, and the circumference of a circle.

### Identifying Polygons

A polygon is a closed, two-dimensional shape classified by the number of straight-line sides. See  for some examples. We show only up to eight-sided polygons, but there are many, many more.
If all the sides of a polygon have equal lengths and all the angles are equal, they are called regular polygons. However, any shape with sides that are line segments can classify as a polygon. For example, the first two shapes, shown in  and , are both pentagons because they each have five sides and five vertices. The third shape  is a hexagon because it has six sides and six vertices. We should note here that the hexagon in  is a concave hexagon, as opposed to the first two shapes, which are convex pentagons. Technically, what makes a polygon concave is having an interior angle that measures greater than . They are hollowed out, or cave in, so to speak. Convex refers to the opposite effect where the shape is rounded out or pushed out.
While there are variations of all polygons, quadrilaterals contain an additional set of figures classified by angles and whether there are one or more pairs of parallel sides. See .

### Perimeter

Perimeter refers to the outside measurements of some area or region given in linear units. For example, to find out how much fencing you would need to enclose your backyard, you will need the perimeter. The general definition of perimeter is the sum of the lengths of the sides of an enclosed region. For some geometric shapes, such as rectangles and circles, we have formulas. For other shapes, it is a matter of just adding up the side lengths.
A rectangle is defined as part of the group known as quadrilaterals, or shapes with four sides. A rectangle has two sets of parallel sides with four angles. To find the perimeter of a rectangle, we use the following formula:
For example, to find the length of a rectangle that has a perimeter of 24 inches and a width of 4 inches, we use the formula. Thus,
     
The length is 8 units.
The perimeter of a regular polygon with  sides is given as . For example, the perimeter of an equilateral triangle, a triangle with three equal sides, and a side length of 7 cm is .

### Sum of Interior and Exterior Angles

To find the sum of the measurements of interior angles of a regular polygon, we have the following formula.
For example, if we want to find the sum of the interior angles in a parallelogram, we have
     
Similarly, to find the sum of the interior angles inside a regular heptagon, we have
     
To find the measure of each interior angle of a regular polygon with  sides, we have the following formula.
For example, find the measure of an interior angle of a regular heptagon, as shown in . We have
     
An exterior angle of a regular polygon is an angle formed by extending a side length beyond the closed figure. The measure of an exterior angle of a regular polygon with  sides is found using the following formula:
In , we have a regular hexagon . By extending the lines of each side, an angle is formed on the exterior of the hexagon at each vertex. The measure of each exterior angle is found using the formula, .
Now, an important point is that the sum of the exterior angles of a regular polygon with  sides equals  This implies that when we multiply the measure of one exterior angle by the number of sides of the regular polygon, we should get  For the example in , we multiply the measure of each exterior angle, , by the number of sides, six. Thus, the sum of the exterior angles is 

### Circles and Circumference

The perimeter of a circle is called the circumference. To find the circumference, we use the formula  where  is the diameter, the distance across the center, or  where  is the radius.
The radius is ½ of the diameter of a circle. The symbol  is the ratio of the circumference to the diameter. Because this ratio is constant, our formula is accurate for any size circle. See .
Let the radius be equal to 3.5 inches. Then, the circumference is
     

### Check Your Understanding


### Key Terms

1. perimeter
2. polygon
3. pentagon
4. hexagon
5. heptagon
6. octagon
7. quadrilateral
8. trapezoid
9. parallelogram
10. circumference

### Key Concepts

1. Regular polygons are closed, two-dimensional figures that have equal side lengths. They are named for the number of their sides.
2. The perimeter of a polygon is the measure of the outline of the shape. We determine a shape’s perimeter by calculating the sum of the lengths of its sides.
3. The sum of the interior angles of a regular polygon with  sides is found using the formula  The measure of a single interior angle of a regular polygon with  sides is determined using the formula 
4. The sum of the exterior angles of a regular polygon is  The measure of a single exterior angle of a regular polygon with  sides is found using the formula 
5. The circumference of a circle is  where  is the radius, or  and  is the diameter.

### Formulas

The formula for the perimeter  of a rectangle is , twice the length  plus twice the width .
The sum of the interior angles of a polygon with  sides is given by
     
The measure of each interior angle of a regular polygon with  sides is given by
     
To find the measure of an exterior angle of a regular polygon with  sides we use the formula
     
The circumference of a circle is found using the formula  where  is the diameter of the circle, or  where  is the radius.