6.4: Polygons and Angles
Introduction
The Geodesic Dome
“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 lesson on polygons and angles. By the end of the lesson, you will be able to help Dylan with his geodesic dome.
What You Will Learn
In this lesson you will learn how to complete the following skills.
- Distinguish among polygons, regular polygons and closed figures that are not polygons, and classify polygons by the number of sides.
- Create or complete a table showing number of sides, diagonals, triangles formed, and sum of angle measures for polygons through decagon.
- Derive and apply the formula for angle measures of regular polygons.
- Describe and classify polygons in real-world objects and architecture.
Teaching Time
I. Distinguish among Polygons, Regular Polygons and Closed Figures that are not Polygons and Classify Polygons by the Number of Sides
In this lesson we will examine polygons. 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.
Here is an example of 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.
This last example is a good example of a type of polygon. 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.
II. Create or Complete a Table Showing Number of Sides, Diagonals, Triangles Formed, and Sum of Angle Measures for Polygons through Decagon
As we just talked about in the last section, 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. Look at this example.
Example
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.
III. Derive and Apply the Formula for Angle Measures of Regular Polygons
In the last section, we used diagonals and triangles to figure out the sum of the interior angles of a polygon. 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*}.
We can use the formula to find the sum of the angles in any polygon. Let’s practice.
Example
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. Then continue with the lesson.
What if the hexagon were a regular hexagon? Then its six angles would have to be congruent. If we know the sum of the angle measures, then we can find the measure of each if we divide the total by 6:
\begin{align*}720^\circ \div 6 = 120^\circ\end{align*}
Every angle in a regular hexagon, no matter how long its sides are, will always be \begin{align*}120^\circ\end{align*}.
This is true for any regular polygon. We can always divide the total number of degrees by the number of angles to find the measure of each angle. This is a way of working backwards. We start with the number of degrees and then divide by the number of sides to figure out each angle measure.
Let’s look at an example.
Example
What is the measure of each angle in a regular octagon?
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*}.
Great job! We can use this method to find the measure of the angles in any regular polygon.
IV. Describe and Classify Polygons in Real – World Objects and Architecture
We can use what we know about the properties of polygons to analyze and classify them in the real world. We still count the number of sides and angles to classify the polygon.
Example
To identify the polygon, count the sides and angles of the building. There are five, so this is a pentagon. In fact, this building is called the Pentagon! The Pentagon is a government office building in Washington, DC. Thousands of government employees work in the building.
Example
To identify the polygon, count the sides and angles in the structure. There are eight, so this is an octagon.
Now that you have become more familiar with the different types of polygons, you will see them everywhere in the world around you!
Real-Life Example Completed
The Geodesic Dome
Here is the original problem once again. Reread it and then use what you have learned from this lesson to write the sum of the angles measures in hexagon and in a pentagon. Remember, there are two parts to your answer.
“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.
Remember, there are two parts to your answer.
Solution to Real – Life Example
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*}
Vocabulary
Here are the vocabulary words found in this lesson. In addition to these words, you need to familiarize yourself with the different types of polygons by knowing the number of sides and angles in each.
- Polygon
- a simple closed figure made up of line segments.
- Regular Polygon
- a polygon with all sides congruent and all angles congruent.
- Irregular Polygon
- a polygon where all sides are not congruent, but all angle measures are congruent.
Time to Practice
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.
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
- Nonagon
- Heptagon
- Pentagon
Directions: Now use the measures that you just found to figure out the measure of each angle in each regular polygon.
- What is the measure of each angle in a regular heptagon?
- A regular decagon?
- A regular octagon?
- A regular hexagon?
- A regular pentagon?