5 Regular Polygons, Tessellations, and Circles
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606
Chapter 12 Geometric Shapes
Vertex angle
Central angle
Exterior angle
Figure 12.69
Notice that the number of vertex angles, central angles, and the sides of a regular polygon are the same. But there are twice as many exterior angles as interior angles. Vertex
angles and exterior angles are formed in all convex shapes whose sides are line segments.
✔
Check for Understanding: Exercise/Problem Set A #1–4
Given that the sum of the measures of the interior angles of a triangle is 180 degrees and the sum of the measures of the angles
around a single point is 360 degrees, find the measure of one interior angle in a regular pentagon.
Write a description of the method you used in a way that could be understood by a peer and could
be applied to other regular polygons. Compare your method with a peer’s.
Angle Measures in Regular Polygons
Figure 12.70
v1
v5
v2
To find the measure of the central angle of a regular n-gon, notice that the sum of the
measures of n central angles is 360°. Figure 12.70 illustrates this when n = 5. Clearly
360°
= 72°.
these central angles are all congruent—the measure of any one of them is
5
360°
In general, the measure of each central angle in a regular n-gon is
.
n
We can find the measure of the vertex angles in a regular n-gon by using the angle
sum in a triangle theorem. Consider a regular pentagon (n = 5; Figure 12.71). Let us
call the vertex angles ∠v1, ∠v2 , ∠v3 , ∠v4 , and ∠v5 . Since all the vertex angles have the
same measure, it suffices to find the vertex angle sum in the regular pentagon. The
measure of each vertex angle, then, is one-fifth of this sum.
To find this sum, subdivide the pentagon into triangles using diagonals AC and
AD (Figure 12.72). For example, A, B, and C are the vertices of a triangle, specifically
nABC . Several new angles are formed, namely ∠a, ∠b, ∠c, ∠d , ∠e, ∠f , and ∠g.
Notice that
v3
v4
m ( ∠v1 ) = m ( ∠a ) + m ( ∠b ) + m ( ∠c ),
m ( ∠v3 ) = m ( ∠d ) + m ( ∠e ),
Figure 12.71
A
and
m ( ∠v4 ) = m ( ∠f ) + m ( ∠g ).
a b c
v2 B
E v5
g
d
f
D
e
C
Figure 12.72
Problem-Solving Strategy
Draw a Picture
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Within each triangle (nABC , nACD, and nADE), we know that the angle sum is
180°. Hence
m( ∠v1 ) + m( ∠v2 ) + m( ∠v3 ) + m( ∠v4 ) + m( ∠v5 )
= m( ∠a ) + m( ∠b ) + m( ∠c ) + m( ∠v2 ) + m( ∠d )
+ m( ∠e ) + m( ∠f ) + m( ∠g ) + m( ∠v5 )
= [ m( ∠c ) + m( ∠v2 ) + m( ∠d )] + [ m( ∠b ) + m( ∠e ) + m( ∠f )]
+ [ m( ∠g ) + m( ∠v5 ) + m( ∠a )]
= 180° + 180° + 180°
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Section 12.5 Regular Polygons, Tessellations, and Circles 607
Since each bracketed sum is the angle sum in a triangle. Hence the angle sum in a
regular pentagon is 3 × 180° = 540°. Finally, the measure of each vertex angle in the
regular pentagon is 540° ÷ 5 = 108°. The technique used here of forming triangles
within the polygon can be used to find the sum of the vertex angles in any polygon.
Table 12.8 suggests a way of computing the measure of a vertex angle in a regular
n-gon, for n = 3, 4, 5, 6, 7, 8. Verify the entries.
TABLE 12.8
n
Vertex
angle
Exterior
angle
Figure 12.73
Algebraic Reasoning
Students often describe the computation of the measures of each
of the angles in this theorem in
words. This is a natural preliminary step to using variables.
ANGLE SUM IN
A REGULAR n-GON
MEASURE OF A
VERTEX ANGLE
3
1 Ơ 180
180 ữ 3 = 60
4
2 Ơ 180
(2 ì 180) ữ 4 = 90
5
3 Ơ 180
(3 ì 180) ữ 5 = 108
6
4 Ơ 180
(4 ì 180) ữ 6 = 120
7
5 Ơ 180
(5 ì 180) ữ 7 = 128 74
8
6 Ơ 180
(6 ì 180) ÷ 8 = 135°
The entries in Table 12.8 suggest a formula for the measure of the vertex angle
in a regular n-gon. In particular, we can subdivide any n-gon into ( n − 2 ) triangles.
Since each triangle has an angle sum of 180°, the angle sum in a regular n-gon is
( n − 2 ) ¥ 180°
. We can also express
( n − 2 ) ¥ 180°. Thus each vertex angle will measure
n
180°n − 360°
360°
this as
= 180° −
. Thus any vertex angle is supplementary to any
n
n
central angle.
To measure the exterior angles in a regular n-gon, notice that the sum of
a vertex angle and an exterior angle will be 180°, by the way the exterior
angle is formed (Figure 12.73). Therefore, each exterior angle will have measure
360° ⎤
360° 360°
⎡
= 180° − 180° +
180° − ⎢180° −
=
. Hence, the measure of any exterior
⎥
n
n
n
⎣
⎦
angle is the same as the measure of a central angle!
We can summarize our results about angle measures in regular polygons as follows.
THEOREM 12.3
Angle Measures in a Regular n-gon
Vertex Angle
Central Angle
Exterior Angle
( n − 2 ) ¥ 180°
n
360°
n
360°
n
Remember that these results hold only for angles in regular polygons—not necessarily in arbitrary polygons. In the problem set, the central angle measure will be used
when discussing rotation symmetry of polygons. We will use the vertex angle measure
in the next section on tessellations.
✔
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Check for Understanding: Exercise/Problem Set A #5–12
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608
Chapter 12 Geometric Shapes
Tessellations
A polygonal region is a polygon together with its interior. An arrangement of
polygonal regions having only sides in common that completely covers the plane
is called a tessellation. We can form tessellations with arbitrary triangles, as Figure
12.74 shows. Pattern (a) shows a tessellation with a scalene right triangle, pattern
(b) a tessellation with an acute isosceles triangle, and pattern (c) a tessellation with
an obtuse scalene triangle. Note that angles measuring x, y, and z meet at each
vertex to form a straight angle. As suggested by Figure 12.74, every triangle will
tessellate the plane.
(a)
x
z y
z
y
x
x
x z y
y
x
(b)
(c)
Figure 12.74
Every quadrilateral will form a tessellation also. Figure 12.75 shows several tessellations with quadrilaterals.
(a)
(b)
(c)
(d)
Figure 12.75
In pattern (a) we see a tessellation with a parallelogram; in pattern (b), a tessellation with a trapezoid; and in pattern (c), a tessellation with a kite. Pattern (d)
shows a tessellation with an arbitrary quadrilateral of no special type. We can form
a tessellation, starting with any quadrilateral, by using the following procedure
(Figure 12.76).
1. Trace the quadrilateral [Figure 12.76(a)].
2. Rotate the quadrilateral 180° around the midpoint of any side. Trace the image
[Figure 12.76(b)].
3. Continue rotating the image 180° around the midpoint of each of its sides, and
trace the new image [Figures 12.76(c) and (d)].
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Section 12.5 Regular Polygons, Tessellations, and Circles 609
d
d
c
b
a
a
a
c b
a
b c
d
d
d
c
a
b c
d
(b)
(a)
d
a
c b
a
b
b
a
c b
b c
a d
d
c
d a
c
b
(d)
(c)
Figure 12.76
The rotation procedure described here can be applied to any triangle and to any
quadrilateral, even nonconvex quadrilaterals.
Several results about triangles and quadrilaterals can be illustrated by tessellations.
In Figure 12.74(c) we see that x + y + z = 180°, a straight angle. In Figure 12.76(d) we
see that a + b + c + d = 360° for the quadrilateral.
✔
Check for Understanding: Exercise/Problem Set A #13–15
When creating a tessellation of a plane, several polygons meet
at each point and the sum of the measures of the interior angles
of these polygons is 360 degrees. For example, the sum of the measures of the interior angles
of three equilateral triangles and two squares is 3 ¥ 60° + 2 ¥ 90° = 360°. These polygons can fit
together in one of two ways as shown. There are 21 different arrangements of regular polygons
that fit around a point. See how many you can find. The two examples of triangles and squares
shown are two of the 21 arrangements.
(4,4,3,3,3)
(4,3,4,3,3)
Tessellations with Regular Polygons
Figure 12.77 shows some tessellations with equilateral triangles, squares, and regular
hexagons. These are examples of tessellations each composed of copies of one regular
polygon. Such tessellations are called regular tessellations.
(a)
(b)
(c)
Figure 12.77
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Chapter 12 Geometric Shapes
TABLE 12.9
n
MEASURE OF VERTEX
ANGLE IN A REGULAR
n-GON
3
60°
4
90°
5
108°
6
120°
7
128 74 °
8
135°
9
140°
10
144°
11
147 113 °
12
150°
Notice that in pattern (a) six equilateral triangles meet at each vertex, in pattern (b)
four squares meet at each vertex, and in pattern (c) three hexagons meet at each vertex. We say that the vertex arrangement in pattern (a)—that is, the configuration of
regular polygons meeting at a vertex—is (3, 3, 3, 3, 3, 3). This sequence of six 3s indicates that six equilateral triangles meet at each vertex. Similarly, the vertex arrangement in pattern (b) is (4, 4, 4, 4), and in pattern (c) it is (6, 6, 6) for three hexagons.
Consider the measures of vertex angles in several regular polygons (Table 12.9). For
a regular polygon to form a tessellation, its vertex angle measure must be a divisor of
360, since a whole number of copies of the polygon must meet at a vertex to form a 360°
angle. Clearly, regular 3-gons (equilateral triangles), 4-gons (squares), and 6-gons (regular hexagons) will work. Their vertex angles measure 60°, 90°, and 120°, respectively,
each measure being a divisor of 360°. For a regular pentagon, the vertex angle measures
108°, and since 108 is not a divisor of 360, we know that regular pentagons will not fit
together without gaps or overlapping. Figure 12.78 illustrates this fact.
108°
108° 108°
36°
Figure 12.78
For regular polygons with more than six sides, the vertex angles are larger than
120° (and less than 180°). At least three regular polygons must meet at each vertex,
yet the vertex angles in such polygons are too large to make exactly 360° with three
or more of them fitting together. Hence we have the following result.
THEOREM 12.4
Tessellations Using Only One Type of Regular n-gon
Only regular 3-gons, 4-gons, or 6-gons form tessellations of the plane by themselves.
If we allow several different regular polygons with sides the same length
to form a tessellation, many other possibilities result, as Figure 12.79 shows.
Notice in Figure 12.79(d) that several different vertex arrangements are possible.
Tessellations such as those in Figure 12.79 appear in patterns for floor and wall
coverings and other symmetrical designs. Tessellations using two or more regular polygons are called semiregular tessellations if their vertex arrangements are
identical. Thus, the tessellation in Figure 12.79(d) is not semiregular, but Figures
12.79(a), (b), and (c) are.
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(4, 8, 8)
(3, 6, 3, 6)
(a)
(b)
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Section 12.5 Regular Polygons, Tessellations, and Circles 611
(3, 3, 3, 3, 3, 3)
(3, 3, 4, 3, 4), (3, 3, 4, 12)
(4, 6, 12)
(d)
(c)
Figure 12.79
✔
Check for Understanding: Exercise/Problem Set A #16–19
Circles
Figure 12.80
If we consider regular n-gons in which n is very large, we can obtain figures with
many vertices, all of which are the same distance from the center. Figure 12.80
shows a regular 24-gon, for example. Now imagine the figure that would result if
you continually increased the number of sides. These figures would become more
and more like a circle. A circle is the set of all points in the plane that are a fixed
distance from a given point (called the center). The distance from the center to a
point on the circle is called the radius of the circle. Any segment whose endpoints
are the center and a point of the circle is also called a radius. The length of a line
segment whose endpoints are on the circle and which contains the center is called a
diameter of the circle. The line segment itself is also called a diameter. Figure 12.81
shows several circles and their centers.
r
Diameter
Radius
Center
Figure 12.81
Figure 12.82
A compass is a useful device for drawing circles with different radii. Figure 12.82
shows how to draw a circle with a compass. We study techniques for constructing
figures with a compass and straightedge in Chapter 14.
If we analyze a circle according to its symmetry properties, we find that it has
infinitely many lines of symmetry. Every line through the center of the circle is a line
of symmetry (Figure 12.83).
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Chapter 12 Geometric Shapes
Mira
Figure 12.83
Figure 12.84
Also, a circle has infinitely many rotation symmetries, since every angle whose vertex
is the center of the circle is an angle of rotation symmetry (Figure 12.84).
Many properties of a circle, including its area, are obtained by comparing the circle
to regular n-gons with increasingly large values of n. We investigate several measurement properties of circles and other curved shapes in Chapter 13.
✔
Check for Understanding: Exercise/Problem Set A #20–21
©Ron Bagwell
In 1994 the World Cup Soccer Championships were held in the
United States. These games were held in various cities and in a variety of stadiums across the country. Unlike American football, soccer
is played almost exclusively on natural grass. This presented a problem for the city of Detroit because its stadium, the Silverdome, is an
indoor field with artificial turf. Growing grass in domed stadiums has
yet to be done with much success, so the organizers turned to the
soil scientists at Michigan State University. They decided to grow the
grass outdoors on large pallets and then move these pallets indoors
in time for the games. The most interesting fact of this endeavor is
the shape of the pallets that they chose—hexagons! Since hexagons
are one of the three regular polygons that form a regular tessellation, these pallets would fit together to cover the stadium floor but
would be less likely to shift than squares or triangles.
EXERCISE/PROBLEM SET A
EXERCISES
1. For each of the following shapes, determine which of the
following descriptions apply.
c.
d.
S : simple closed curve
C : convex, simple closed curve
N : n-gon
a.
b.
2. Use your protractor to measure each vertex angle in each of
the following polygons. Extend the sides of the polygon, if
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Section 12.5 Regular Polygons, Tessellations, and Circles 613
necessary. Then find the sum of the measures of the vertex
angles. What should the sum be in each case?
a.
b.
5. Find the missing angle measures in each of the following
quadrilaterals.
a.
b.
c.
6. For the following regular n-gons, give the measure of a
vertex angle, a central angle, and an exterior angle.
a. 12-gon b. 16-gon c. 10-gon d. 20-gon
3. Using correct notation, identify three exterior angles and
two vertex angles in pentagon BDGJL.
C
A
10. Given are the measures of the exterior angles of regular
polygons. How many sides does each one have?
a. 9° b. 45° c. 10°
L
K
G
J
F
H
I
4. Draw the lines of symmetry in the following regular
n-gons. How many does each have?
a.
b.
c.
8. Given the following measures of a vertex angle of a regular polygon, determine how many sides each one has.
a. 140° b. 162° c. 178°
9. Given are the measures of the central angles of regular
polygons. How many sides does each one have?
a. 30° b. 72° c. 5°
E
D
B
7. The sum of the measures of the vertex angles of a certain
polygon is 3420°. How many sides does the polygon have?
d.
11. Given are the measures of the vertex angles of regular polygons. What is the measure of the central angle of each one?
a. 90° b. 176° c. 150°
12. Given are the measures of the exterior angles of regular polygons. What is the measure of the vertex angle of each one?
a. 72° b. 10° c. 2°
13. On a square lattice, draw a tessellation with each of the
following triangles. You may find the Chapter 12 eManipulative Geoboard on our Web site to be helpful in thinking
about this problem.
a.
b.
14. Given is a portion of a tessellation based on a scalene triangle. The angles are labeled from the basic tile.
e. This illustrates that a regular n-gon has how many lines
of symmetry?
f. If n is odd, each line of symmetry goes through a _____
and the _____ of the opposite side.
g. If n is even, half of the lines of symmetry connect a
_____ to the opposite _____. The other half connect the
_____ of one side to the _____ of the opposite side.
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Chapter 12 Geometric Shapes
a. Are lines l1 and l2 parallel?
b. What does the tessellation illustrate about corresponding angles?
17. The dual of a tessellation is formed by connecting the
centers of polygons that share a common side. The dual
tessellation of the equilateral triangle tessellation is shown.
Find the dual of the other tessellations.
c. What is illustrated about alternate interior angles?
d. Angle 1 is an interior angle on the right of the transversal. Angles 2 and 3 together form the other interior
angle on the right of the transversal. From the tessellation, what is true about m( ∠1) + [ m( ∠2 ) + m( ∠3)] ? This
result suggests that two lines are parallel if and only if
the interior angles on the same side of a transversal are
_____ angles.
15. Illustrated is a tessellation based on a scalene triangle
with sides a, b, and c. The two shaded triangles are similar
(have the same shape). For each of the corresponding
three sides, find the ratio of the length of one side of the
smaller triangle to the length of the corresponding side
of the larger triangle. What do you observe about corresponding sides of similar triangles?
Complete the following statements.
a. The dual of the regular tessellation with triangles is a
regular tessellation with _____.
b. The dual of the regular tessellation with squares is a
regular tessellation with _____.
c. The dual of the regular tessellation with hexagons is a
regular tessellation with _____.
18. A tessellation is a semiregular tessellation if it is made
with regular polygons such that each vertex is surrounded
by the same arrangement of polygons.
16. a. Given are portions of the (3, 3, 3, 3, 3, 3), (4, 4, 4, 4),
and (6, 6, 6) tessellations. In the first, we have selected
a vertex point and then connected the midpoints of the
sides of polygons meeting at that vertex. The resulting
figure is called the vertex figure. Draw the vertex figure
for each of the other tessellations.
a. One of these arrangements was (3, 3, 4, 12), as shown.
Can point B be surrounded by the same arrangement of
polygons as point A? What happens to the arrangement
at point C ?
b. Can the arrangement (3, 3, 4, 12) be extended to form a
semiregular tessellation? Explain.
c. Find another arrangement of regular polygons that fit
around a single point but cannot be extended to a semiregular tessellation.
19. Shown are copies of an equilateral triangle, a square, a regular hexagon, a regular octagon, and a regular dodecagon.
b. A tessellation is a regular tessellation if it is constructed
of regular polygons and has vertex figures that are regular polygons. Which of the preceding tessellations are
regular?
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Section 12.5 Regular Polygons, Tessellations, and Circles 615
a. Label the measure of one vertex angle for each
polygon.
b. Use the Chapter 12 eManipulative activity Tessellations
on our Web site to find the number of ways you
can combine three of these figures (they may be repeated) to surround a point without gaps and
overlaps.
c. By sketching, record each way you found.
20. Paper folding can be used to find the diameter and center
of a circle.
21. In the circle below, O is the center. What kind of triangle
is nAOB? Explain.
Diameter: Fold the paper so one-half of the circle
exactly lines up with the other half. This fold line will
be the diameter.
A
Center: The center is found by using paper folding to
find a second diameter. The center is the point where
the two diameters intersect.
O
Trace the following circle onto a piece of paper and use
paper folding to determine if the segment and point in the
circle are the diameter and center.
B
PROBLEMS
22. a. Given a square and a circle, draw an example where
they intersect in exactly the number of points given.
i. No points
ii. One point
iii. Two points
iv. Three points
b. What is the greatest number of possible points of intersection?
23. Explain how the shaded portion of the tessellation
illustrates the Pythagorean theorem for isosceles right
triangles.
25. Complete the following table. Let V represent the number
of vertices, D the number of diagonals from each vertex,
and T the total number of diagonals.
POLYGON
V
D
T
Triangle
Quadrilateral
Pentagon
Hexagon
⋅⋅⋅
Octagon
n-gon
24. Calculate the measure of each lettered angle. Congruent
angles and right angles are indicated.
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26. Suppose that there are 20 people in a meeting room. If
every person in the room shakes hands once with every
other person in the room, how many handshakes will
there be?
27. In the five-pointed star that follows, what is the sum of
the angle measures at A, B, C , D, and E ? Assume that the
pentagon is regular.
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Chapter 12 Geometric Shapes
28. A man observed the semiregular floor tiling shown here
and concluded after studying it that each angle of a
regular octagon measures 135°. What was his possible
reasoning?
30. Redo Problem 27 by constructing a five-pointed star on
the Geometer’s Sketchpad®. Measure the angles at the
points of the star and add them up by using the Measure
and Calculate options in the software. Moving the vertices
of the star will change some of the angle measures.
a. What do you observe about the sum of the measured
angles?
b. Justify your observation from part a.
31. Trace the following hexagon twice.
29. Find the maximum number of points of intersection for
the following figures. Assume that no two sides coincide
exactly.
a. A triangle and a square
b. A triangle and a hexagon
c. A square and a pentagon
d. An n-gon ( n > 2 ) and a p-gon ( p > 2 )
a. Divide one hexagon into three identical parts so that
each part is a rhombus.
b. Divide the other hexagon into six identical kites.
EXERCISE/PROBLEM SET B
EXERCISES
1. For each of the following shapes, determine which of the
following descriptions apply.
S : simple closed curve
C : convex, simple closed curve
N : n-gon
a.
2. Use your protractor to measure each vertex angle
in each polygon. Extend the sides of the polygon
shown if necessary. Then find the sum of the measures
of the vertex angles. What should the sum be in
each case?
a.
b.
b.
c.
d.
3. Using correct notation, identify two exterior angles, two
vertex angles, and two central angles in the hexagon
GHIJKL.
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