Have you ever heard of a geodesic dome? Take a look at this dilemma.

“I am going to design a house that no one has ever even thought of before,” Dylan said during Mrs. Patterson’s class on Tuesday.

“What do you mean?” Kelsey inquired.

“A dome made of triangles. How’s that for an idea!” Dylan said grinning from ear to ear.

“It’s great,” Kelsey agreed, “But it already exists. It is called a “geodesic dome”.”

Dylan looked at Kelsey as she pulled open a book and showed Dylan the exact page where there was information written on the geodesic dome. He shrugged his shoulders.

“Well, I am going to do it anyway,” he said.

Dylan began to explore the geodesic dome. He figured out that the average dome has isosceles triangles in it. These triangles, like all triangles, have a sum of \begin{align*}180^\circ\end{align*} for angle measures. Dylan wants to construct a geodesic dome.

As he begins to draw his design, he figures out that he will need a variety of hexagons and pentagons. Dylan is stuck. He can’t figure out how many degrees there will be in each hexagon and how many degrees there will be in each pentagon.

**This is where you come in. Pay attention to this Concept on polygons. By the end of it, you will be able to help Dylan with his geodesic dome.**

### Guidance

*Polygons***are two-dimensional figures that have three or more sides. Any figure with straight edges, such as a triangle or rectangle, is a polygon.** If a figure has any curved sides or is open, it is not a polygon.

This is a figure that is not a polygon.

Polygons have special properties that determine their angle and side relationships. For instance, the number of sides a polygon has is related to the number of angles it has, and therefore determines the sum of its angles.

Now that we can distinguish polygons from other figures, let’s take a closer look at them. In general, there are two kinds of polygons: regular polygons and irregular polygons.

*Regular polygons***have sides and angles that are all congruent.** It doesn’t matter how many sides and angles they have. As long as the sides are congruent and the angles are congruent, the figure is a regular polygon.

*Irregular polygons***, you may have guessed, are polygons that do** *not***have congruent sides and angles.** They are still polygons because they have straight, closed sides. Their sides are simply different lengths.

**Here are two hexagons. The first one is a regular hexagon. You can see that the side lengths and the angle measures are all congruent. The second figure is still a hexagon, but it is an irregular hexagon. While it has six sides, it has different side lengths and angle measures.**

While we can have different triangles and different quadrilaterals, we can also have different types of polygons besides regular and irregular. We can further identify polygons according to the number of sides that they have. The hexagon (6 sides) is an example of one of these types of polygons. Other examples include: triangles, pentagons, octagons and decagons. Let’s look at the different types of polygons in more detail.

We can distinguish between different types of polygons according to the number of sides that each has. This is how we can name the polygon. We can also look at different characteristics of each type of polygon. We can look at the number of sides, the number of diagonals that can be drawn in a figure, the number of triangles in the polygon and the sum of the angle measures.

The easiest way to approach this is through the use of a table. Let’s begin with naming polygons, looking at images of polygons, examining the number of angles and sides and the sum of the interior (inside) angles.

Polygon Name |
Polygon |
Number of Angles and Sides |
Sum of Interior Angles |
---|---|---|---|

triangle | 3 | @$\begin{align*}180^\circ\end{align*}@$ | |

rectangle/square | 4 | @$\begin{align*}360^\circ\end{align*}@$ | |

pentagon | 5 | @$\begin{align*}540^\circ\end{align*}@$ | |

hexagon | 6 | @$\begin{align*}720^\circ\end{align*}@$ | |

heptagon | 7 | @$\begin{align*}900^\circ\end{align*}@$ | |

octagon | 8 | @$\begin{align*}1,080^\circ\end{align*}@$ | |

nonagon | 9 | @$\begin{align*}1,260^\circ\end{align*}@$ | |

decagon | 10 | @$\begin{align*}1,440^\circ\end{align*}@$ |

You can see that polygons have similar names. **In the word** *polygon, poly***-means “many” and-***gon***means “angle.” So** *polygon***means “having many angles.”** Now look at the name for the shape that has eight angles and sides. It is called an *octagon*. In *octagon*, *oct*-means “eight.” An octopus, for example, has eight arms. In *pentagon*, *pent*-means “five,” so this is a shape with five angles and sides.

**One thing to notice about these polygons is that they can all be divided by diagonals. We can figure out how many diagonals there are and by doing this divide them up into triangles. We know that the sum of the angle measures of a triangle is @$\begin{align*}180^\circ\end{align*}@$, and we can use this information to figure out the sum of the angle measures of the different polygons.**

Take a look at this situation.

**Notice that this hexagon has been divided using diagonals. There are three diagonals in the hexagon which create four triangles. Each of these triangles has @$\begin{align*}180^\circ\end{align*}@$ in it, so we can multiply @$\begin{align*}180 \times 4\end{align*}@$ to find the sum of the degrees inside a hexagon.**

@$\begin{align*}180 \times 4 = 720^\circ\end{align*}@$

**This is the answer and you can see how this corresponds to the number of degrees in the chart.**

**We can apply this information to any of the polygons. Simply divide the polygon into triangles and multiply the number of triangles by 180.**

**We can also use a formula to find the sum of a polygon’s interior angles.** Knowing the total is helpful because we often can use it to find the measure of a particular angle in the polygon. Remember, in a regular polygon, all of the angles are congruent. We can find the angle of all of them if we know the total and how many angles there are.

**As we have seen, we can find the total number of degrees in a polygon by using triangles. The formula sums this up nicely and gives us a shortcut:**

@$\begin{align*}(n - 2) \times 180^\circ\end{align*}@$

The letter @$\begin{align*}n\end{align*}@$ represents the number of angles (or sides) in the polygon. In other words, we subtract 2 from the number of angles and then multiply by 180. Think about the different polygons, the number of triangles in a polygon is always 2 less than the number of sides. The formula is simply giving us a shortcut to find the number of triangles in the polygon. Then, as we know, we multiply by @$\begin{align*}180^\circ\end{align*}@$.

**Find the sum of the angles a hexagon.**

First, count the number of angles or sides. This polygon has six sides and six angles. We will put 6 in for @$\begin{align*}n\end{align*}@$ in the formula and solve.

@$$\begin{align*}(n - 2) &\times 180^\circ\\ (6 - 2) &\times 180^\circ\\ 4 \times 180^\circ &= 720^\circ\end{align*}@$$

**The formula tells us that a hexagon contains 4 triangles. When we multiply by @$\begin{align*}180^\circ\end{align*}@$, we find that the sum of the interior angles in a hexagon is @$\begin{align*}720^\circ\end{align*}@$. This is true for any hexagon, regular or irregular.**

**Write this formula down in your notebook.**

Figure out the sum of the angle measures in each polygon.

#### Example A

Nonagon

**Solution: @$\begin{align*}1260^\circ\end{align*}@$**

#### Example B

Heptagon

**Solution: @$\begin{align*}900^\circ\end{align*}@$**

#### Example C

Pentagon

**Solution: @$\begin{align*}540^\circ\end{align*}@$**

Now let's go back to the dilemma from the beginning of the Concept.

**Now let’s break down the solution to the problem. Let’s start with hexagons.**

We know that there are six triangles in a hexagon. We know that the sum of the angle measures of a triangle is @$\begin{align*}180^\circ\end{align*}@$. However, we have to take the number of sides into consideration. We can use the following formula to help us. The letter @$\begin{align*}n\end{align*}@$ represents the number of sides.

@$$\begin{align*}(n - 2) &\times 180\\ (6 - 2) &\times 180 = 720^\circ\end{align*}@$$

**Now let’s look at the pentagon.** The pentagon is comprised of 5 triangles. We know that the sum of the angle measures of each triangle is @$\begin{align*}180^\circ\end{align*}@$. We can use the same formula as we did with the hexagon.

@$$\begin{align*}(n - 2) &\times 180\\ (5 - 2) &\times 180 = 540^\circ\end{align*}@$$

### Guided Practice

Here is one for you to try on your own.

What is the measure of each angle in a regular octagon?

**Solution**

If it is a regular octagon, all of the angles are congruent. We need to find the total number of degrees in an octagon and then divide by 8, because an octagon has 8 angles. Let’s use the formula to find the total of the angles.

@$$\begin{align*}&(n - 2) \times 180^\circ\\ &(8 - 2) \times 180^\circ\\ &6 \times 180^\circ\\ &1,080^\circ\end{align*}@$$

The 8 angles in an octagon must have a sum of @$\begin{align*}1,080^\circ\end{align*}@$. Check your table to be sure. Now that we know the total, we divide by 8 to find the measure of each angle.

@$\begin{align*}1,080^\circ \div 8 = 135^\circ\end{align*}@$

**Each angle in a regular octagon, no matter how big or small, always measures @$\begin{align*}135^\circ\end{align*}@$.**

### Video Review

### Explore More

Directions: Answer true or false to each question about regular and irregular polygons.

- The angles of a regular polygon are all the same size.
- A regular hexagon has six sides that are different lengths.
- An irregular pentagon has sides that are the same length.
- An irregular polygon is one that has one side open.
- A regular triangle could also be called an equilateral triangle.
- The side lengths of a regular octagon are all the same length.

Directions: Identify each figure as regular or irregular. Then identify the type of polygon that it is too.

- An eight sided figure with sides of equal length.

Directions: Use the formula @$\begin{align*}(n - 2) \times 180\end{align*}@$ to figure out the sum of the angle measures of each polygon.

- Regular hexagon
- Regular octagon
- Triangle
- Trapezoid
- Decagon