8.4: Polygons
Introduction
The Hexagon
“Wow!” Juanita exclaimed, looking at a glass structure outside the art museum. “Come look at this,” she called to her friend Samantha.
Samantha came over to see what Juanita was so excited about.
“What’s going on?” Samantha asked.
“Look at all of the figures in this sculpture,” Juanita commented. “There are so many different ones. I can see squares, trapezoids and triangles.”
“Oh, yeah, I can see those too,” Samantha agreed. “Look, I can also see a hexagon, and it is made up of 6 equilateral triangles. That means that the sum of all of the angles of the hexagon is \begin{align*}720^\circ\end{align*}.”
“How do you figure that?” Juanita asked shaking her head.
“Just look at it. You see those triangles, well if you add up the contributions from the two base angles of each triangle you get a sum of \begin{align*}1080^\circ\end{align*}. Oh, I can’t explain it,” Samantha said seeing Juanita’s confused look.
Can you explain it? To explain Samantha’s theory, you have to understand polygons and their angles. This lesson will teach you everything that you need to know to work through this problem. Take a few notes as you go through this lesson. At the end of it, you will need what you have learned to solve this problem.
What You Will Learn
By the end of this lesson you will be able to complete the following tasks:
- Recognize and classify quadrilaterals.
- Recognize and classify polygons.
- Recognize regular polygons as having all sides and angles congruent.
- Describe and analyze polygons and associated angle measures using known classification and sufficient given information.
Teaching Time
I. Recognize and Classify Quadrilaterals
In this lesson we will examine figures called polygons. Polygons are closed shapes with sides made up of lines. Any shape with straight edges, such as a triangle or rectangle, is a polygon. We will learn to identify each kind of polygon. Let’s take a look at some polygons.
Polygons with four sides, such as rectangles and squares, are called quadrilaterals. Quadrilaterals have special properties.
What is a quadrilateral?
The prefix of this word “quad” means four. Therefore, a quadrilateral is a polygon that has four sides. They also have four angles, and these four angles always have a sum of \begin{align*}360^\circ\end{align*}. This is true no matter what shape or size the quadrilateral is. Take a look at the quadrilaterals below.
Each figure has four sides and four angles. Notice how different the angles and the sides of the quadrilaterals are, though. Now look closely. If you add up the measures of the four angles in any of the quadrilaterals, they always equal \begin{align*}360^\circ\end{align*}!
Another important feature of quadrilaterals is that many have one or even two pairs of parallel sides. Look at these quadrilaterals again.
Can you find the pairs of parallel sides?
We can also classify quadrilaterals even further. You can see that each of the figures above has four sides and four angles, but each is different based on the lengths of the sides, the angle measures and the types of sides.
Let’s look at the different types of quadrilaterals in detail.
A parallelogram is a quadrilateral with opposite sides parallel. The first figure above is a parallelogram. There are a few special features of parallelograms. One is that each pair of parallel sides is congruent. In the first figure above, the two short sides are the same length and the two long sides are the same length. This special relationship between the pairs of sides affects the angles of a parallelogram. This gives parallelograms their other distinguishing feature: the angles opposite each other are also congruent. The \begin{align*}70^\circ\end{align*} angles are opposite each other, and the \begin{align*}110^\circ\end{align*} angles are opposite each other. This relationship exists in any parallelogram, no matter the length of the sides or the sizes of the angle pairs.
There are three special kinds of parallelograms: rectangles, squares, and a rhombus.
A rectangle is a quadrilateral because it has four sides, and it is a parallelogram because it has two pairs of parallel, congruent sides. Now take a look at its angles. All four angles are right angles! Therefore a rectangle is any shape with two pairs of parallel sides and four right angles (bear in mind that the pairs of angles opposite each other are still equal!).
A square also has two pairs of parallel sides and four right angles. It is special, though, because all four of its sides are congruent.
The third special parallelogram is called a rhombus. Like a square, a rhombus has four congruent sides. It does not have right angles, but it does still have pairs of congruent angles opposite each other. So a rhombus can be a square, but a square isn’t necessarily a rhombus because of the right angles necessary in a square.
Let’s try recognizing and classifying some parallelograms.
Example
Identify the shapes below as a rectangle, square, rhombus, or parallelogram.
We know that all parallelograms have two pairs of parallel sides. To distinguish them, we need to analyze the angles and compare the lengths of each pair of sides.
The first figure does not have right angles, so it cannot be a rectangle or square. Now compare the sides. One pair is 5 centimeters and the other pair is also 5 centimeters. Four congruent sides without right angles make this a rhombus.
Figure 2 does have right angles, so it must be either a rectangle or a square. Compare the pairs of sides to find out. One pair is 2 inches, but the other pair is only 1 inch. This figure does not have four congruent sides, so it is a rectangle.
Now let’s look at Figure 3. No right angles here. What about the sides? The pairs are not the same length, so it cannot be a rhombus. This is a parallelogram; it does not have any other special features.
The next figure does. It has four right angles and four congruent sides, so it is a square.
Now let’s look at one other special quadrilateral. A trapezoid is a quadrilateral that has only one pair of parallel sides.
Keep the characteristics of each type of quadrilateral in mind as you identify different four sided figures!
Here is a chart with all that you need to identify the different quadrilaterals.
We’ve learned to identify all the special kinds of quadrilaterals. Remember also that no matter how long a quadrilateral’s sides are, or whether or not any sides are parallel, their four angles always add up to \begin{align*}360^\circ\end{align*}.
8H. Lesson Exercises
Identify each type of quadrilateral based on its description.
- A four sided figure with opposite sides parallel.
- A four sided figure with opposite sides parallel and congruent with four right angles.
- A four sided figure with one pair of parallel sides. Opposite sides are congruent.
Check your answers with a friend.
II. Recognize and Classify Polygons
In the beginning of this lesson we mentioned the word “polygon.” A polygon is a closed figure made up of lines and angles. Now we are going to look at polygons in more detail.
We classify polygons by the number of sides and the number of angles in them.
Here is a table with information on some of the different types of polygons.
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*} |
Write down each polygon in the chart, its number of sides and the sum of its interior angles.
Look at the table.
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.
Now take a look at the column on the right. Each kind of polygon has a different sum of its interior angles. For instance, the three angles in a triangle always add up to \begin{align*}180^\circ\end{align*}, and the four angles in a quadrilateral always add up to \begin{align*}360^\circ\end{align*}. No matter how long or short the sides of a triangle are, the angles must total \begin{align*}180^\circ\end{align*}. We can also identify polygons by the total number of degrees of their interior angles.
Let’s practice classifying some polygons.
Example
Identify each polygon below.
Count the number of angles or sides. The first figure has six angles and sides. Check the table. Six angles and sides make it a hexagon.
You may already recognize the next figure. A triangle is a polygon that has three angles and sides.
Figure 3 is more unusual. It has nine angles and sides. This means it is a nonagon. Non-means “nine.”
The next figure has five angles and sides. Look at the table. A polygon having five angles and sides is called a pentagon.
You may recognize this shape too. It has the same number of sides as a rectangle, but it isn't quite one because all rectangles have opposite sides that are parallel. All rectangles are four-sided polygons. And, as we have learned, they are also called quadrilaterals. Quad means “four.”
Count the number of angles or sides in the last figure. It has seven angles and sides. This makes it a heptagon. Hept-means “seven.”
Sometimes, we can use polygons when problem solving. Let’s look at this example.
Example
Morgan stitched a geometric figure into part of a quilt she is sewing. She measured the angles to help her know where to stitch. The sum of the angle measures was \begin{align*}1,080^\circ\end{align*}. What kind of polygon did she stitch?
In this example, we have no easy way of knowing how many angles or sides the figure has. What do we know? We know that the sum of the interior angles, however many there are, is \begin{align*}1,080^\circ\end{align*}. Only one polygon has angles that always add up to this amount. Check the table above.
Morgan must have stitched an octagon. Now we know that the distinguishing properties of an octagon are not only that it has eight sides and angles, but that its eight angles must have a sum of \begin{align*}1,080^\circ\end{align*}.
8I. Lesson Exercises
Use what you have learned to identify each polygon described:
- The sum of the interior angles is \begin{align*}180^\circ\end{align*}.
- It has seven sides.
- It has five sides and five angles.
Check your work with a friend and then move on to the next section.
III. Recognize Regular Polygons as Having All Sides and Angles Congruent
A polygon can have a certain number of sides, but the sides do not necessarily have to be the same length. Both of the polygons below are pentagons because they both have five angles and sides. Look how different they are, though!
Count the angles or sides. Each has five, so each is definitely a pentagon.
What makes them look so different?
Look at the lengths of the sides. In the first pentagon, all of the angles are congruent and all of the sides are congruent. In the second pentagon, the angles have different measures and the sides have different lengths.
We call a polygon whose angles and sides are all congruent a regular polygon. Any polygon can be a regular polygon.
Look at the regular polygons below.
As you can see, a regular hexagon has six congruent sides. It also has six congruent angles. Remember that the sum of a hexagon’s angles is always \begin{align*}720^\circ\end{align*}. Because the six angles in a regular hexagon are congruent, they are all \begin{align*}720 \div 6 = 120^\circ\end{align*}. A regular hexagon, no matter how long its sides, will always have angles that measure \begin{align*}120^\circ\end{align*}.
A regular octagon has eight congruent angles and sides. It doesn’t matter how long the sides are, as long as they are congruent. Every regular octagon has the same angles. Can you find the measure of a regular octagon’s eight angles?
Well, we know that the sum of the interior angles in an octagon is \begin{align*}1,080^\circ\end{align*}. Because the angles are congruent, we divide by 8 to find the measure of each:
\begin{align*}1,080 \div 8 = 135^\circ\end{align*}
Every regular octagon has angles of \begin{align*}135^\circ\end{align*}.
We don’t usually say “regular rectangle,” because a rectangle with congruent sides is actually a square. As we know, the four angles in a square are always \begin{align*}90^\circ\end{align*}, and their sum is always \begin{align*}360^\circ\end{align*}. We also have a special name for a “regular triangle.” A triangle with three congruent sides is called an equilateral triangle. Its three angles are always \begin{align*}60^\circ\end{align*}, and their sum is always \begin{align*}180^\circ\end{align*}.
Any polygon that does not have all congruent sides is an irregular polygon. Irregular polygons can still be pentagons, hexagons, and nonagons, but they do not have congruent angles or equal sides. Here are some examples of irregular polygons.
In these polygons, some sides are clearly longer than others. Some angles are wider than others. They cannot be regular polygons if their angles are different measures or their sides are different lengths, so they must be irregular. Notice that irregular polygons tend to look uneven or lopsided, while regular polygons look more orderly and symmetrical.
Let’s practice identifying irregular and regular polygons.
Example
Identify each polygon below and tell whether it is regular or irregular.
First, count the number of angles or sides to find which kind of polygon it is.
Figure 1 has six angles and sides, so it is a hexagon. Do all the sides appear to be the same length? If you’re not sure, use a ruler to check. They are all equal, so this is a regular hexagon.
Figure 2 has seven angles and sides, so it is a heptagon. Notice how short the top side is compared to the bottom side. These sides are definitely not the same length, so this cannot be a regular polygon. Therefore it is an irregular heptagon.
The next figure has five angles and sides, so it is a pentagon. The angles are all \begin{align*}108^\circ\end{align*}, so this is a regular pentagon because it has congruent angles.
How many sides does this polygon have? It has eight, which makes it an octagon. It looks a bit lopsided, so it’s likely that the sides are not all congruent. Let’s take a look at the angles. Some of them are the same measure, but not all of them. It cannot be a regular polygon because all of its angles are not congruent; it is an irregular octagon.
We can identify a polygon by the number of sides and angles it has, or by the sum of its interior angles. Then we can tell whether it is regular or irregular depending on whether all the angles or sides are congruent.
IV. Describe and Analyze Polygons and Associated Angle Measures using Classification and Given Information
Now that you know about polygons, we can use this information to analyze polygons and their relationships.
When lines intersect in the geometric plane, they form polygons. We can apply what we know to these polygons in order to classify them, find side lengths, or solve for unknown angle measures. Take a look at the diagram below.
The intersection of these five lines has created several different polygons. First, let’s see if we can find them all.
The largest is \begin{align*}AMDE\end{align*}. Within this figure there are two figures. What are they?
One is figure \begin{align*}ABCDE\end{align*}. What can we tell about figure \begin{align*}ABCDE\end{align*}? First, we can determine what kind of polygon it is. It has five angles and five sides, so it is a pentagon. Two of its angles measure \begin{align*}90^\circ\end{align*}, and one measures \begin{align*}120^\circ\end{align*}. Can we find the measures of the two remaining angles? We can, but to do so, we need to take a look at the third figure.
The third figure is triangle \begin{align*}MBC\end{align*}. It has two \begin{align*}60^\circ\end{align*} angles. We can use these angles to help us find the measures of the unknown angles in the pentagon \begin{align*}ABCDE\end{align*}. Angles \begin{align*}ABC\end{align*} and \begin{align*}MBC\end{align*} are supplementary. In other words, together they form a straight line. A straight line measures \begin{align*}180^\circ\end{align*}. Therefore the sum of these two angles is \begin{align*}180^\circ\end{align*}. We simply subtract to find the measure of \begin{align*}ABC\end{align*}.
\begin{align*}180 - 60 = 120^\circ\end{align*}
Angle \begin{align*}ABC\end{align*} is \begin{align*}120^\circ\end{align*}. Draw a copy of the diagram and fill this information in. Now let’s see if we can find the measure of angle \begin{align*}BCD\end{align*}. We now know four of the five angles in figure \begin{align*}ABCDE\end{align*}. Because this is a pentagon, we also know that its interior angles must have a sum of \begin{align*}540^\circ\end{align*}. We can set up an equation to find the measure of the unknown angle.
\begin{align*}90 + 90 + 120 + 120 + \angle BCD &= 540^\circ\\ 420 + \angle BCD &= 540^\circ\\ \angle BCD &= 540 - 420\\ \angle BCD &= 120^\circ\end{align*} The fifth and final angle must measure \begin{align*}120^\circ\end{align*}. Let’s add up all of the angles in the pentagon to be sure they total \begin{align*}540^\circ\end{align*}.
\begin{align*}90 + 90 + 120 + 120 + 120 = 540^\circ\end{align*}
We still have one unknown angle, angle \begin{align*}AMD\end{align*}. Can we find its measure? In fact, we have two different ways! It is the third angle in triangle \begin{align*}BMD\end{align*}, and we know that the three angles in a triangle have a sum of \begin{align*}180^\circ\end{align*}. It is also the fourth angle in quadrilateral \begin{align*}AMDE\end{align*}, and we know that the four angles in a quadrilateral have a sum of \begin{align*}360^\circ\end{align*}. We can set up an equation to find \begin{align*}AMD\end{align*}.
\begin{align*}60 + 60 + \angle AMD &= 180^\circ\\ 120 + \angle AMD &= 180^\circ\\ \angle AMD &= 180 - 120\\ \angle AMD &= 60^\circ\end{align*}
We have used the properties of polygons to find all of the unknown angles in the diagram. Now that we know all of the angles, let’s classify each polygon as regular or irregular.
What about quadrilateral \begin{align*}AMDE\end{align*}?
It has two right angles, but the other two are \begin{align*}120^\circ\end{align*} and \begin{align*}60^\circ\end{align*}. Therefore it is irregular.
Pentagon \begin{align*}ABCDE\end{align*} also appears to have sides of different lengths; let’s check the angle measures to be sure. Two angles measure \begin{align*}90^\circ\end{align*} and three measure \begin{align*}120^\circ\end{align*}. The angles are not all congruent, so this cannot be regular either.
What about triangle \begin{align*}BMD\end{align*}? Each of the three angles measures \begin{align*}60^\circ\end{align*}. This is an equilateral triangle, so it is definitely a regular polygon.
Great work! Now let’s apply what we have learned to the problem from the introduction!
Real-Life Example Completed
The Hexagon
Here is the original problem once again. Reread it and underline any important information.
“Wow!” Juanita exclaimed, looking at a glass structure outside the art museum. “Come look at this,” she called to her friend Samantha.
Samantha came over to see what Juanita was so excited about.
“What’s going on?” Samantha asked.
“Look at all of the figures in this sculpture,” Juanita commented. “There are so many different ones. I can see squares, trapezoids and triangles.”
“Oh, yeah, I can see those too,” Samantha agreed. “Look, I can also see a hexagon, and it is made up of 6 equilateral triangles. That means that the sum of all of the angles of the hexagon is \begin{align*}720^\circ\end{align*}.”
“How do you figure that?” Juanita asked shaking her head.
“Just look at it. You see those triangles, well if you add up the contributions from the two base angles of each triangle you get a sum of \begin{align*}1080^\circ\end{align*}. Oh, I can’t explain it,” Samantha said seeing Juanita’s confused look.
You have explored the content of this Concept, and you are now able to write a short explanation or equation that shows how someone might figure out that the sum of the angles of a hexagon is equal to \begin{align*}720^\circ\end{align*}. Take a few minutes to do that now.
Next, let’s look at a possible way of understanding the sum of these angles.
We know that the sum of the angles of a triangle is equal to \begin{align*}180^\circ\end{align*}. If there are six equilateral triangles in a hexagon, and each triangle is \begin{align*}180^\circ\end{align*}, then we can write this equation.
\begin{align*}The \ number \ of \ degrees \ in \ a \ hexagon &= 6 (180)\\ We \ only \ count \ the \ angles \ on \ the \ border \\ of \ the \ figure \ so \ each \ corner \ contributes \ 120^\circ \\ The \ number \ of \ degrees \ in \ a \ hexagon &= 720^\circ\end{align*}
Take this one step further by examining the other polygons in the figure. Make a note of the places where you see squares, trapezoids and triangles.===Vocabulary===
Here are the vocabulary words that are found in this lesson.
- Polygon
- simple closed figure made up of straight lines and angles. Polygons are identified by the number of sides and angles in them.
- Quadrilateral
- a four sided figure
- Parallelogram
- a quadrilateral with opposite sides parallel.
- Rectangle
- a parallelogram with opposite sides congruent, parallel and with four right angles.
- Square
- a rectangle with four congruent sides.
- Rhombus
- a parallelogram with four congruent sides.
- Regular Polygon
- Side lengths and angle measures are congruent.
- Irregular Polygon
- Side lengths and angle measures are not congruent.
Technology Integration
Khan Academy Sum of Interior Angles of a Polygon
James Sousa, Introduction to Polygons
James Sousa, Classifying Polygons
James Sousa, Angles of Convex Polygons
Other Videos:
- http://www.mathplayground.com/mv_polygon_angle_sum.html – This is a Brightstorm video on polygons, their sides and the sum of their angles.
- http://www.math-videos-online.com/polygon-names.html – This is a very cute animated video on classifying polygons.
Time to Practice
Directions: Identify each quadrilateral as a parallelogram, rectangle, square, rhombus, or trapezoid.
1.
2.
3.
4.
Directions: Identify each polygon and tell whether it is regular or irregular.
Directions: Identify each quadrilateral by its description.
13. A quadrilateral with one pair of parallel sides.
14. A quadrilateral with opposites sides congruent and parallel, with four right angles.
15. A quadrilateral with four congruent parallel sides and four right angles.
16. A rectangle with four congruent parallel sides.
Directions: Identify the polygons in the diagram. Then find the measures of the unknown angles.
17.
Directions: Answer true or false for each of the following questions.
18. A rhombus is always a square.
19. A parallelogram has opposite sides that are parallel.
20. A rectangle is a type of parallelogram.
21. Squares, rectangles and rhombi are parallelograms with four right angles.
22. A trapezoid has four right angles.
23. A trapezoid has one pair of parallel sides.
24. A regular polygon has congruent sides and angles.
25. A pentagon can not be an irregular polygon.