8.10: Polygon Classification
“Look," said Samantha as she and Juanita looked at the glass sculpture. "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*}1080^\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 them up 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 Concept will teach you everything that you need to know to work through this problem. Take a few notes as you go through the Concept. At the end of it, you will need what you have learned to solve this problem.
Guidance
In the last Concept, 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 it.
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.
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’s a rectangle. 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.”
Use what you have learned to identify each polygon described.
Example A
The sum of the interior angles is \begin{align*}180^\circ\end{align*}.
Solution: Triangle
Example B
It has seven sides.
Solution: Heptagon
Example C
It has five sides and five angles.
Solution: Pentagon
Vocabulary
Here are the vocabulary words in this Concept.
- 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.
Guided Practice
Here is one for you to try on your own.
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?
Answer
In this dilemma, 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*}.
Video Review
Here is a video for review.
- This is a James Sousa video on classifying polygons.
Practice
Directions: Identify the polygons in the diagram. Then find the measures of the unknown angles.
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Directions: Answer true or false for each of the following questions.
2. A rhombus is always a square.
3. A parallelogram has opposite sides that are parallel.
4. A rectangle is a type of parallelogram.
5. Squares, rectangles and rhombi are parallelograms with four right angles.
6. A trapezoid has four right angles.
7. A trapezoid has one pair of parallel sides.
Directions: Determine whether or not each image is a polygon. If yes, write polygon, if no, write not a polygon.
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15. A square tile on a floor
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diagonal (of a polygon)
A non-side line segment that connect two vertices of a convex polygon.Polygon
A polygon is a simple closed figure with at least three straight sides.Concave
A concave polygon has at least one interior angle greater than 180 degrees. A common way to identify a concave polygon is to look for a "caved-in" side of the polygon.Convex
A convex polygon contains no interior angles greater than 180 degrees.Diagonal
A diagonal is a line segment in a polygon that connects nonconsecutive verticesExterior angles
An exterior angle is the angle formed by one side of a polygon and the extension of the adjacent side.Interior angles
Interior angles are the angles inside a figure.Regular Polygon
A regular polygon is a polygon with all sides the same length and all angles the same measure.Vertices
Vertices are points where line segments intersect.Equilateral
A polygon is equilateral if all of its sides are the same length.Image Attributions
Here you'll learn to recognize and classify polygons.