594
9.28
Chapter 9
Geometric Figures
A few regular polygons are shown in Figure 9.28.
Equilateral
triangle
Figure 9.28
E X AMPLE C
Square
Regular
pentagon
Regular
hexagon
Regular
heptagon
The following figures satisfy only one of the two conditions for regular polygons. For each
polygon determine which condition is satisfied and which condition is not satisfied.
1.
2.
Rhombus
Hexagon
Solution 1. The four sides are congruent, but the four angles are not congruent. 2. The six
angles are congruent, but the six sides are not congruent.
DRAWING REGULAR POLYGONS
There are three special angles in regular polygons (see Figure 9.29). A vertex angle is
formed by two adjacent sides of the polygon; a central angle is formed by connecting the
center of the polygon to two adjacent vertices of the polygon; and an exterior angle is
formed by one side of the polygon and the extension of an adjacent side.
Figure 9.29
Vertex angle
Central angle
Exterior angle
The sum of the measures of the angles in a polygon can be used to compute the
number of degrees in each vertex angle of a regular polygon: Simply divide the sum of all
the measures of the angles by the number of angles. For example, Figure 9.30a on the
next page shows a regular pentagon that is subdivided into three triangles. Since each
vertex of each triangle is a vertex of the pentagon, the sum of the nine angles in the triangles equals the sum of the five angles in the pentagon. So the sum of the angles in the
pentagon is 3 3 1808, or 5408. Therefore, each angle in a regular pentagon is 5408 4 5,
or 1088, as shown in Figure 9.30b.
Section 9.2
1
9
Polygons and Tessellations
595
108°
8
2
7
3
108°
108°
6
4
Figure 9.30
9.29
5
108°
108°
(a)
(b)
Figure 9.31 shows the first four steps for drawing a regular pentagon. The process
begins in step 1, where a line segment is drawn and a point for the vertex of the angle is
marked. Then in step 2 the baseline of a protractor is placed on the line segment so that the
center of the protractor’s baseline is at the vertex point, and a 1088 angle is drawn. In step 3
two sides of the pentagon are marked off, and in step 4 the protractor is used to draw another
1088 angle. This process can be continued to obtain a regular pentagon.
10
0 1
0
100 90 80 70
60
12
50
10
Step (1) Draw a line segment and mark a vertex.
20
170 1
60
15
30
0
14
40
0
13
Step (2) Measure off a 108° angle.
170 160 150 1
40
13
0
12
0
0
11
0
10
90
80 70 60 5
0
40
30
20
108°
10
108°
Figure 9.31
Step (3) Mark off two sides of equal length.
Step (4) Measure off a second angle of 108°.
Another approach to drawing regular polygons begins with a circle and uses central
angles. The number of degrees in the central angle of a regular polygon is 360 divided by
the number of sides in the polygon. A decagon has 10 sides, so each central angle is
3608 4 10, or 368 (Figure 9.32).
36°
Figure 9.32
Central
angle
596
9.30
Chapter 9
Geometric Figures
A four-step sequence for drawing a regular decagon is illustrated in Figure 9.33. The
first and third steps use a compass, a device for drawing circles and arcs and marking off
equal lengths. The decagon that is obtained is said to be inscribed in the circle. Any polygon whose vertices are points of a circle is called an inscribed polygon.
36°
Step (1) Draw a
circle with a compass.
Figure 9.33
Step (2) Measure
a 36° angle with a
protractor.
Step (3) Mark 10
equal lengths with
a compass.
Step (4) Connect
the points to form
a decagon.
TESSELLATIONS WITH POLYGONS
The hexagonal cells of a honeycomb provide another example of regular polygons in nature
(Figure 9.34). The cells in this photograph show that regular hexagons can be placed side
by side with no uncovered gaps between them. Any arrangement in which nonoverlapping
figures are placed together to entirely cover a region is called a tessellation. Floors and
ceilings are often tessellated, or tiled, with square-shaped material, because squares can be
joined together without gaps or overlaps. Equilateral triangles are also commonly used for
tessellations. These three types of polygons—regular hexagons, squares, and equilateral
triangles—are the only regular polygons that will tessellate.
Figure 9.34
Honeycomb with bees