Definition 2.4.1.
A polygon is a regular polygon if all of its sides and all of its angles are congruent.
Section 2.4 Exploring Regular Polygons
Have you ever studied a tiling of a floor or a pattern on a quilt? In this section, we will define a regular polygon and explore when regular tilings can be used to tile the plane.
Subsection 2.4.1 Regular Polygons
We begin by defining what it means for a polygon to be regular.
Checkpoint 2.4.2. Check Your Understanding!
Explain why each shape below is not a regular polygon:
Subsection 2.4.2 Regular Tilings
In Exploration 1.1.1, we created a tiling of a square using tetrominoes. In this section, we will consider tilings of the plane. Since the plane continues infinitely, we will not be able to create a complete tiling. However, we will be able to build enough of the tiling to determine whether a tiling is possible.
Definition 2.4.4.
A tiling is a collection of closed polygons that cover a plane (continuing infinitely in all directions) with no gaps and no overlaps. A monohedral tiling is a tiling that uses only one shaped tile while a regular tiling is a monohedral tiling that uses a single regular polygon as its tile. In a regular tiling, each side of a polygon will line up perfectly with a side of an adjacent c.
Exploration 2.4.1. Regular Tilings.
This activity may be completed using physical tiles or in the interactive applet below. Be sure to save a record of your designs by tracing, photography, or screenshots. Some of the computations will also be used later in this section, so you are encouraged to keep an accessible record of your work for future reference.
The web application Polypad by Amplify provides regular \(n\)-gons for \(3\leq n\leq 8\text{.}\) These are the top six shapes provided. Simply drag the shape from the left into the whiteboard area and it will make a new copy. Rotate the polygon using the dotted stem and move it by clicking on the interior and dragging. For \(n\)-gons with \(n>8\text{,}\) use the nonagon at the bottom. Pull on the dotted vertex to change the number of sides. With the new \(n\)-gon selected, you can now use the duplicated sheet icon on the flip/cut/... menu to make copies of your \(n\)-gon. Fullscreen, undo, redo, reset, and extract buttons are provided on the right.
1
mathigon.org/polypad#polygons2
mathigon.org/polypad/2RtlO1JO7dpDA(a)
Select a single regular polygon, make copies of it as needed, and use it to construct a regular tiling. Construct enough of the tiling to convince others that the tiling can continue to tile the plane, upward and downward, to the left and the right, with no gaps and overlaps.
(b)
Experiment with equilateral triangles, squares, regular pentagons, regular hexagons, and other regular \(n\)-gons to determine which can be used to create a regular tiling and which cannot. What do you think will happen with regular polygons with more than eight sides? Why?
(c)
Perhaps you noticed that the vertex angle measure plays a role in determining whether a regular polygon can be used to create a regular tiling. The steps in this task will lead you to determine the measure of the vertex angle of a regular pentagon. We will use this technique to find the measure of the vertex angles for other regular \(n\)-gons.
The following shorthand will be used to refer to the angles.
- \(\displaystyle \angle 1 = \angle EAD\)
- \(\displaystyle \angle 2 = \angle DAC\)
- \(\displaystyle \angle 3 = \angle CAB\)
- \(\displaystyle \angle 4 = \angle EDA\)
- \(\displaystyle \angle 5 = \angle ADC\)
- \(\displaystyle \angle 6 = \angle DCA\)
- \(\displaystyle \angle 7 = \angle ACB\)
- \(\angle E = \angle DEA\text{,}\) etc.
- Explain why \(m\angle EAB=m\angle 1+m\angle 2+m\angle 3\)
- What is \(m\angle 1 + m\angle 4 + m\angle E\text{?}\)
- Use the fact that the pentagon is divided into triangles to determine \(m\angle EAB+m\angle B+m\angle BCD+m\angle CDE +m\angle E\text{.}\)
- Having just computed the sum of the five congruent angles of a regular pentagon, determine \(m\angle A\text{.}\)
- Some of what you found, but not all, will be true for all convex pentagons. What results about angle measure will also hold for non-regular convex pentagons? What will be different?
(d)
Use your work from the previous task to complete the row for regular pentagon in the table below. Repeat this procedure for squares, regular hexagons, regular octagons, and regular decagons: First draw diagonals from a single vertex to dissect the polygon into triangles, next use the triangles to determine the total vertex angle measure, and then determine the measure of each angle. In the final row, \(n\) is a variable representing the number of sides and the answers will be algebraic expressions involving the variable \(n\text{.}\)
| shape | number of sides |
number of triangles |
total angle measure |
single vertex measure |
|---|---|---|---|---|
| triangle | 3 | 1 | 180 | 60 |
| square | 4 | |||
| regular pentagon | 5 | |||
| regular hexagon | 6 | |||
| regular octagon | 8 | |||
| regular decagon | 10 | |||
| regular \(n\)-gon | \(n\) |
(e)
Returning to the question of when a regular polygon can be used to create a regular tiling,
- How many equilateral triangles can meet at a single point without overlapping?
- How many squares can meet at a single point?
- How many regular hexagons can meet at a single point?
- Explain in words and pictures why these three regular polygons can each produce a regular tiling.
- When we tried to surround a point with regular pentagons a gap was formed. What is the degree measure of that gap?
- If \(n\geq 7\text{,}\) do you know whether a regular \(n\)-gon might produce a regular tiling? Include a discussion of vertex angle measure in your explanation.
In Exploration 2.4.1, we determined the measure of a vertex angle of a regular \(n\)-gon. We define this term and another important angle.
Definition 2.4.8.
A vertex angle of a polygon is any angle which shares its vertex with that of the polygon. The sides of a vertex angle lie along the two sides of the polygon, meeting at the vertex.
When a polygon is regular, we can also define its center to be the point which is equidistant from all vertices of the polygon. A central angle of a regular polygon has its vertex at the center of the polygon and its sides consist of a pair of rays emanating from the center to two adjacent vertices.
Subsection 2.4.3 Semiregular Tilings
If we allow more than one regular polygon to be used in a tiling, we are able to create a wider variety of designs.
Definition 2.4.9.
A semiregular tiling is a tiling consisting of two or more regular polygons with common side lengths with the additional requirement that the arrangement of polygonal faces around every vertex is the same.
We can denote the vertex arrangement by identifying the number of sides in each polygon as we travel around a vertex. An example of this is the 3.3.4.3.4-vertex arrangement, referring to triangle, triangle, square, triangle, square. This arrangement is shown in Figure 2.4.10 and will be discussed in the next exploration.
Exploration 2.4.2. In Search of Semiregular Tilings.
Once again, you are encouraged to continue the practice of saving your sketches, work, and results as they may be useful later in this section.
(a)
What must be true about the measures of the angles of the vertices meeting at and surrounding a single point?
(b)
In Table 2.4.7, you determined the vertex angle measure for a few regular polygons and developed a general formula for finding this measure. Use the formula, to determine the vertex angle measure of each of the following regular \(n\)-gons. If the answer is not a whole number, write the answer as a mixed number.
- heptagon (7 sides)
- nonagon (9 sides)
- dodecagon (12 sides)
- pentakaidecagon (15 sides)
- octakaidecagon (18 sides)
- icosagon (20 sides)
- tetrakaicosaigon (24 sides)
- tetracontakaidigon (42 sides)
Hint.
The measure of the vertex angle of a regular \(n\)-gon is given by the formula \(d=\frac{180(n-2)}{n}\text{.}\)
(c)
Refer to Figure 2.4.10 as you answer the following:
- Verify that the sum of the angles meeting at the vertex is 360 degrees.
- Extend the \(3.3.4.3.4\) design making sure that you always have one or two triangles between any two squares. Will this eventually fill the plane without gaps or overlaps?
- Note that \(3.3.3.4.4\) gives a different pattern. Show that it is possible to create a tiling using the vertex arrangement \(3.3.3.4.4\) by building outward from your initial vertex arrangement.
(d)
Verify arithmetically that it is possible for an equilateral triangle, a regular heptagon, and a regular tetracontakaidigon (pictured below), to fit snugly around a vertex with no overlaps. In shorthand, we can write this vertex arrangement as \(3.7.42\text{.}\)
(e)
Use the angle measures you found in Table 2.4.7 and Task 2.4.2.b to list all combinations of three, four, or five regular polygons that could meet at a vertex, writing the vertex configurations in shorthand notation. Like the example in Task 2.4.2.c, the same shape may be repeated. If more than three polygons meet at a vertex, consider the different ways in which they could be arranged.
Hint.
We are not including the three regular tilings in this list. There is one more vertex arrangement using five polygons (in addition to 3.3.4.3.4 and 3.3.3.4.4), five using four polygons, and eight more using three polygons (in addition to 3.7.42).
(f)
Some of the vertex arrangements in the previous task will extend to create a semiregular tiling, but not all of them! Experiment by surrounding one vertex using the pattern and then try to surround the neighboring vertices using the same arrangement. Which seem to work and which fail? For the ones that fail, what seems to go wrong?
Hint.
Only eight of the seventeen possible vertex arrangements can be used to create a semiregular tiling.
(g)
What do you notice about the vertex arrangements that work (or those that don’t work)? What do you wonder?
This activity may have involved more of a struggle than some of the others. Perhaps you only found about half of the possibilities in Task 2.4.2.e or you cannot yet see a pattern to help determine which vertex arrangements lead to a semiregular tiling. That is perfectly normal. Write down the ideas that you do have and share them with classmates. As you share what you have observed and listen to others’ discoveries, you and your classmates may uncover some amazing results.
Subsection 2.4.4 Narrowing Down the Options
When we look at possible vertex arrangements for a semiregular tiling, our options are very limited. Not including the regular tiling by squares, you should have found three groups of four regular polygons with an angle sum of 360 degrees. All of these have at least one polygon appearing twice. Why do some of these extend to a semiregular tiling while others do not? Does it matter whether the common polygons are next to each other or separated?
As we attempt to create a 3.3.6.6 pattern, we find that it is impossible to maintain a consistent vertex arrangement. Fairly quickly, we are forced to create a 3.6.3.6 arrangement at some vertices. On the other hand, a 3.3.6.6 vertex arrangement can continue indefinitely. Let us explore why this happens.
Exploration 2.4.3. Semiregular Tilings with Four Regular Polygons at a Vertex.
(a)
Use physical or virtual triangles and hexagons to sketch the vertex arrangement 3.3.6.6 or refer to an earlier sketch if available. The Polypad by Amplify applet in Figure 2.4.5 may be used.
- Choose one of the two triangles at this vertex and attempt to surround it with triangles and/or hexagons while maintaining the 3.3.6.6 arrangement at each vertex.
-
Note that each triangle in a 3.3.6.6 must share a side with one triangle and one hexagon due to the order of the 3’s and 6’s.
Figure 2.4.12. Surrounding a regular triangle with regular triangles and regular hexagons. Will the remaining side abut a triangle or a hexagon? Discuss what happens in each case.
(b)
There is a second group of four regular polygons consisting of two equilateral triangles and two other polygons.
- What are the other two polygons? We will represent the number of sides of these polygons by \(m\) and \(n\text{.}\)
- Explain why the arrangement \(3.3.m.n\) will not extend to a semiregular tiling using an argument similar to what we did with \(3.3.6.6\text{.}\)
- Explain why the arrangement \(3.m.3.n\) also fails to create a semiregular tiling.
(c)
The remaining semiregular tiling with four polygons meeting at each vertex has two squares and two other polygons in the vertex arrangement.
- What are the other two polygons? Again, the number of sides of these two polygons are represented by \(m\) and \(n\) in the questions that follow.
- Are you able to surround a square along its sides by an alternating sequence of \(m\)-gons and \(n\)-gons?
- One of the polygons has an odd number, \(m\text{,}\) of sides. Are you able to surround the \(m\)-gon along its sides by an alternating sequence of squares and \(n\)-gons?
- Explain in words and pictures why \(m.4.n.4\) extends to a semiregular tiling, but \(m.4.4.n\) does not. Draw or record a sketch of \(m.4.n.4\) if you do not already have one.
We conclude that there are only two semiregular tilings and one regular tiling of the plane where four polygons meet at a vertex. All three vertex arrangements with five polygons meeting extend to semiregular tilings. To complete this analysis, we consider which three-polygon vertex arrangements extend to a semiregular tiling.
Exploration 2.4.4. Semiregular Tilings with Three Regular Polygons at a Vertex.
(a)
Not including the regular tiling \(6.6.6\text{,}\) there are nine possible vertex arrangements consisting of three regular polygons. List as many as you can, referring back to the work you did in Task 2.4.2.e.
(b)
Verify algebraically that \(5.5.10\) can surround a single point.
(c)
Use the fact that 5 is an odd number to explain why \(5.5.10\) does not extend to a semiregular tiling. You may include pictures as well as words in your explanation.
Hint.
What happens as you try to surround a pentagon with the necessary pentagons and decagons?
(d)
Determine the three vertex arrangements with three polygons that extend to a semiregular tiling. Why must the others be eliminated?
Exercises 2.4.5 Exercises
Building Our Toolbox
1.
In your own words and pictures, describe the difference between a regular and a nonregular polygon.
2.
If a group of polygons meet at a single vertex, what must be the sum of the angle measures at that vertex? Note that this wording generalizes this fact to nonregular polygons as well.
3.
Describe the three regular tilings.
4.
What is the sum of the measures of the vertex angles for any (not necessarily regular) convex \(n\)-gon? For example, the measures of the angles of any triangle add up to 180 degrees. What is the total for a convex quadrilateral, a convex pentagon, etc.?
5.
What formula can be used to compute the angle measure for a regular \(n\)-gon?
6.
What do we mean by a \(m.n.p\)-vertex arrangement?
7.
Just because a collection of regular polygons can surround a vertex does not mean that they can create a semiregular tiling. Give an example of a vertex arrangement that does not extend to a regular tiling. Support your claim.
Skills and Recall
8.
Sketch a picture of each of the following:
- A concave octagon
- A convex hexagon that is not regular
- A rectangle that is a regular quadrilateral.
- A rectangle that is not a regular quadrilateral.
9.
What is the total angle measure of each of the following?
- a triangle
- a convex quadrilateral
- a convex 11-gon
- a convex 50-gon
10.
Demonstrate that each of the following exists by sketching an example:
- a pentagon that is not convex
- a convex hexagon with at least one right angle
11.
Demonstrate that each of the following exists by sketching an example:
- an equilateral polygon that is not regular
- an equiangular polygon that is not regular
12.
Answer the following:
- What is the measure of each vertex in a regular 18-gon?
- What is the measure of each vertex in a regular 9-gon?
- If a regular 18-gon, a regular 9-gon, and an equilateral triangle meet at a vertex, will there be a gap? an overlap? or will they fit together without gaps or overlaps?
- Explain why it is impossible to create a semiregular tiling using regular 18-gons, regular 9-gons, and equilateral triangles.
13.
Answer the following questions about a 100-gon.
- Diagonals drawn from a single vertex will divide a 100-gon into how many triangles?
- What is the sum of the interior angles in a convex 100-gon?
- In a regular 100-gon, what is the measure of each interior angle?
- In a regular 100-gon, what is the measure of each central angle?
Hint.
What must the sum of the central angles be?
14.
In Figure 2.4.13, \(\angle{EAB}\) and \(\angle{ABC}\) are right angles, \(m(\angle{CDE})=99^\circ\text{,}\) and \(\angle{AED}\cong\angle{BCD}\text{.}\) Determine \(m(\angle{AED})\text{.}\)
15.
Explain why each of the following are not semiregular tilings:
- The design started in Figure 2.4.14.
- The design started in Figure 2.4.15
- The design started in Figure 2.4.16
Extending the Concepts
16. Another Way to Calculate the Vertex Angle of a Regular Polygon.
In Figure 2.4.17, a regular nonagon, its center \(O\text{,}\) and segments joining each vertex to the center are shown. Use it to answer the following:
- Explain why each central angle must measure 40 degrees.
- What type of triangle is \(\Delta AOB\) ? Why?
- What is the measure of \(\angle{ABO}\) ? Why?
- What is the measure of \(\angle{ABC}\) ? Explain based on the steps in this problem.
- What is the formula for the central angle of any regular \(n\)-gon?
- Explain how we can use the measure of a central angle to justify that the vertex angle of a regular \(n\)-gon has measure \(180-\frac{360}{n}\text{.}\)
- Algebraically, show that the vertex angle of a regular \(n\)-gon has measure \(\frac{180(n-2)}{n}\) using the steps above.
17. Finding the Center of a Regular Polygon.
In Exercise 2.4.5.16, the center of the regular nonagon was provided. In this problem, we explore how to find the center of a regular \(n\)-gon.
- Give a definition for the center of a regular polygon in your own words.
- First give clear instructions on how one can find the center of any regular \(n\)-gon where \(n\) is an even number. Explain your technique. You may use the decagon in Figure 2.4.18 for exploration and illustration. This applet will also allow you to create other \(n\)-gons as needed. The “circle and radius tool” may be used to check your work, but not for finding the center.
- Refer to Figure 2.4.17. Do lines connecting vertices pass through the center? Give an alternate strategy for drawing lines that will pass through the center of an \(n\)-gon when \(n\) is odd. You may use the nonagon in the GeoGebra applet in Figure 2.4.18. The “circle and radius tool” may be used to check your work, but not for finding the center.
Writing Prompts
18.
Write a letter to a friend describing one thing you found to be surprising or interesting in this section.
19.
Write a letter to a friend in which you give a formula for the vertex angle of a regular \(n\)-gon and explain why it holds for all regular \(n\)-gons.
20.
Write a letter to a friend in which you explain why there are exactly three regular tilings.
