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2. 9.1 ? Points, Lines, Planes, and AnglesOne of the first steps in learning geometry is learning the terms and definitions of geometry. This section covers these terms and definitions.. 3. Euclidean Geometry - History:In approx 300 B.C., Euclid summarized much of the Greek mathematics of his ti
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1. 1 Chapter 9 – Geometry
9.1 Points, Lines, Planes, and Angles
9.2 Polygons
9.3 Perimeter and Area
9.4 Volume
9.5 The Möbius Strip, Klein Bottle, and Maps
9.6 Non-Euclidean Geometry
2. 2 9.1 – Points, Lines, Planes, and Angles
One of the first steps in learning geometry is learning the terms and definitions of geometry. This section covers these terms and definitions.
3. 3 Euclidean Geometry - History:
In approx 300 B.C., Euclid summarized much of the Greek mathematics of his time. He summarized it in his greatest work, The Elements. The Elements was a 13 book set which laid the foundation for plane geometry, called Euclidean geometry.
Interestingly, these books were studied by Abraham Lincoln when he was in law school.
4. 4 Euclidean Geometry - History:
Euclid was the first mathematician of his time to use the axiomatic method.
Euclid’s axiomatic system consisted of four parts: undefined terms, which lead to definitions, which lead to postulates (also called axioms, which are accepted as true), which lead to theorems (which are proven by deductive reasoning).
5. 5 Euclidean Geometry – History:
The undefined terms on which Euclid based his system were point, line, and plane.
Point – a location in space
Line – a straight arrangement of points
Plane – a two dimensional surface that extends infinitely in both directions ( i.e., a table top )
Euclid used the undefined terms to introduce certain definitions as they were needed in his axiomatic system.
6. 6 Euclidean Geometry – Definitions:
half line – a piece of a line containing all the points lying to one side of a given endpoint.
ray – a piece of a line which contains an endpoint and all the points lying to one side of the endpoint.
line segment – a piece of a line with two endpoints
Notice how in the definition of half line, an endpoint is not included. These, as well as open line segment and half open line segment, are illustrated in Figure 9.2 on page 432 in the text. Note the diagram and the symbols for the different terms.
7. 7 Euclidean Geometry – Definitions:
Remember intersection and union? These are used again in geometry. Look at Example 1 on page 433-434 in the text. This example illustrates how to determine the solution to different intersections and unions of lines, rays, half lines, etc. Pay special attention to the symbols, especially the half line.
8. 8 Euclidean Geometry – Definitions:
Remember that a plane is a two dimensional surface that extends infinitely in both directions. Some properties of planes can be found in Figures 9.4 and 9.5 on page 434 in the text. Planes are very important to some of Euclid’s definitions.
parallel lines – are two lines, lying in a plane that do not intersect.
parallel planes – are two planes that remain the same distance apart.
skew lines – are two lines that do not intersect and do not lie on the same plane.
9. 9 Euclidean Geometry – Definitions:
angle – the union of two rays with a common endpoint (? ).
vertex – the common endpoint of the two rays.
sides of an angle – the rays that make up an angle.
More about angles can be found in Figure 9.6 on page 434 in the text. Example 2 on page 435 of the text illustrates how to find unions and intersections that involve angles and rays.
10. 10 Euclidean Geometry – Definitions:
measure of an angle – the size of an angle measured by the amount of rotation needed to turn one side of the angle to the other by pivoting about the vertex.
Angles are measured in degrees ( ? ).
protractor – a device used to measure angles.
11. 11 Euclidean Geometry – Definitions:
right angle – an angle that measures 90?.
acute angle – an angle that measures less than 90?.
obtuse angle – an angle that measures more than 90?, but less than 180?.
straight angle – an angle that measures 180?.
12. 12 Euclidean Geometry – Definitions:
adjacent angles – two angles that have a common vertex and side, but no common interior points (i.e.,?1&?2, ?2&?3,? 3&?4, ?4&?1).
vertical angles – a pair of nonadjacent angles formed by two intersecting lines (i.e., ?1&?3 , ?2&?4). Vertical angles are congruent in measure.
13. 13 Euclidean Geometry – Definitions:
complementary angles – two angles whose sum measures 90?.
supplementary angles – two angles whose sum measures 180?.
14. 14 Euclidean Geometry – Definitions:
transversal – a line that intersects two different lines at two different points.
When two parallel lines are cut by a transversal, eight angles are formed. (lines m & n, below, are parallel)
15. 15 Euclidean Geometry – Definitions:
When two parallel lines are cut by a transversal, interior and exterior angles are formed.
16. 16 Euclidean Geometry – Definitions:
When two parallel lines are cut by a transversal, interior and exterior angles are formed.
interior angles – angles that lie between the parallel lines (?3,?4,?5,&?6).
17. 17 Euclidean Geometry – Definitions:
When two parallel lines are cut by a transversal, interior and exterior angles are formed.
interior angles – angles that lie between the parallel lines (?3,?4,?5,&?6).
exterior angles – angles that lie outside the parallel lines (?1,? 2,?7,& ?8).
18. 18 Euclidean Geometry – Definitions:
alternate interior angles (AIA) – interior angles on opposite sides of the transversal.
alternate exterior angles (AEA) – exterior angles on opposite sides of the transversal.
corresponding angles (CA) - one interior and one exterior angle on the same side of a transversal. They are located in the same place in each group of four angles.
When two parallel lines are cut by a transversal, AIA, AEA, and CA are congruent in measure.
19. 19 Euclidean Geometry – Definitions:
20. 20 Euclidean Geometry – Definitions:
Alternate interior angles, alternate exterior angles, and corresponding angles are also summarized at the bottom of page 437 in the text.
An important thing to remember is that alternate interior angles, alternate exterior angles, and corresponding angles are congruent (the same measure) only when the two lines are parallel. If you are not told that the lines are parallel, then the measures of the angles may not be congruent.
21. 21 Angle Diagram #1:
22. 22 Angle Diagram #1:
23. 23 Angle Diagram #1:
24. 24 Angle Diagram #1:
25. 25 Angle Diagram #1:
26. 26 Angle Diagram #1:
27. 27 Angle Diagram #1:
28. 28 Angle Diagram #1:
29. 29 Angle Diagram #1:
30. 30 Angle Diagram #1:
31. 31 Angle Diagram #1:
32. 32 Angle Diagram #2:
33. 33 Angle Diagram #2:
34. 34 Angle Diagram #2:
35. 35 Angle Diagram #2:
36. 36 Angle Diagram #2:
37. 37 Modeling Angles #3:
The difference between the measures of two complementary angles is 24?. Determine the measures of the two angles.
38. 38 Modeling Angles #3:
The difference between the measures of two complementary angles is 24?. Determine the measures of the two angles.
Define the variables:
let x = the larger angle
let y = the smaller angle
39. 39 Modeling Angles #3:
The difference between the measures of two complementary angles is 24?. Determine the measures of the two angles.
Define the variables:
let x = the larger angle
let y = the smaller angle
Write the 1st equation:
x + y = 90
Sums to 90 since the two angles are complementary.
40. 40 Modeling Angles #3:
Write the 2nd equation:
x – y = 24
The difference of the two angles is 24?.
41. 41 Modeling Angles #3:
Write the 2nd equation:
x – y = 24
The difference of the two angles is 24?.
The system of equations:
x + y = 90
x – y = 24
42. 42 Modeling Angles #3:
Write the 2nd equation:
x – y = 24
Since the difference of the two angles is 24?.
The system of equations:
x + y = 90
x – y = 24
2x = 114
43. 43 Modeling Angles #3:
Write the 2nd equation:
x – y = 24
Since the difference of the two angles is 24?.
The system of equations:
x + y = 90
x – y = 24
2x = 114
x = 57
44. 44 Modeling Angles #3:
Write the 2nd equation:
x – y = 24
Since the difference of the two angles is 24?.
The system of equations:
x + y = 90
x – y = 24
2x = 114
x = 57
45. 45 Modeling Angles #3:
Write the 2nd equation:
x – y = 24
Since the difference of the two angles is 24?.
The system of equations:
x + y = 90
x – y = 24
2x = 114
x = 57
46. 46 Modeling Angles #3:
Write the 2nd equation:
x – y = 24
Since the difference of the two angles is 24?.
The system of equations:
x + y = 90
x – y = 24
2x = 114
x = 57
47. 47 Modeling Angles #4:
If ?1 & ?2 are supplementary angles and ?1 is eight times as large as ?2,find the measures of ?1 & ?2.
48. 48 Modeling Angles #4:
If ?1 & ?2 are supplementary angles and ?1 is eight times as large as ?2,find the measures of ?1 & ?2.
Define the variables:
let x = measure ?2
let 8x = measure ?1
49. 49 Modeling Angles #4:
If ?1 & ?2 are supplementary angles and ?1 is eight times as large as ?2,find the measures of ?1 & ?2.
Define the variables:
let x = measure ?2
let 8x = measure ?1
Write the equation:
8x + x = 180
50. 50 Modeling Angles #4:
If ?1 & ?2 are supplementary angles and ?1 is eight times as large as ?2,find the measures of ?1 & ?2.
Define the variables:
let x = measure ?2
let 8x = measure ?1
Write the equation:
8x + x = 180
Solve the equation:
9x = 180
x = 20
51. 51 Modeling Angles #4:
If ?1 & ?2 are supplementary angles and ?1 is eight times as large as ?2,find the measures of ?1 & ?2.
Define the variables:
let x = measure ?2
let 8x = measure ?1
Write the equation:
8x + x = 180
Solve the equation:
9x = 180
x = 20
52. 52 9.2 - Polygons
This section covers polygons, which are figures that are made up of line segments. Many of the theorems in geometry have to do with polygons.
53. 53 Polygon Definitions:
polygon – a closed planar figure created by three or more straight segments.
sides of a polygon – the straight segments that form a polygon.
vertices of a polygon (singular, vertex) – the points where two sides meet in a polygon.
polygonal region – the union of the sides of a polygon and its interior.
54. 54 Polygon Definitions:
equilateral polygon – a polygon whose sides are the same length.
equiangular polygon – a polygon whose angles are equal in measure.
regular polygon – a polygon whose sides are the same length and whose angles are equal in measure.
55. 55 Polygon Definitions:
Polygons are named according to their number of sides. Table 9.1 on page 441 in the text lists the names of different polygons, and their corresponding number of sides.
56. 56 Triangles:
One of the most important polygons is the triangle. An important thing to know about triangles is that the sum of the measures of the interior angles of a triangle is 180 degrees.
57. 57 Angle Diagram - Triangles:
58. 58 Angle Diagram - Triangles:
59. 59 Angle Diagram - Triangles:
60. 60 Angle Diagram - Triangles:
61. 61 Angle Diagram - Triangles:
62. 62 Angle Diagram - Triangles:
63. 63 Angle Diagram - Triangles:
64. 64 Angle Diagram - Triangles:
65. 65 Angle Diagram - Triangles:
66. 66 Angle Diagram - Triangles:
67. 67 Angle Diagram - Triangles:
68. 68 Angle Diagram - Triangles:
69. 69 Angle Diagram - Triangles:
70. 70 Angle Diagram - Triangles:
71. 71 Angle Diagram - Triangles:
72. 72 Angle Diagram - Triangles:
73. 73 Angle Diagram - Triangles:
74. 74 Angle Diagram - Triangles:
75. 75 Angle Diagram - Triangles:
76. 76 Angle Diagram - Triangles:
77. 77 Angle Diagram - Triangles:
78. 78 Interior angles of a polygon:
The sum of the measures of the interior angles of an n-sided polygon is (n-2)180.
This formula is based on the idea that the interior angles of a triangle sum to 180?.
79. 79 Interior angles of a polygon:
The sum of the measures of the interior angles of an n-sided polygon is (n-2)180.
This formula is based on the idea that the interior angles of a triangle sum to 180?.
Given a pentagon:
80. 80 Interior angles of a polygon:
The sum of the measures of the interior angles of an n-sided polygon is (n-2)180.
This formula is based on the idea that the interior angles of a triangle sum to 180?.
Given a pentagon:
Draw in the diagonals from one vertex.
81. 81 Interior angles of a polygon:
The sum of the measures of the interior angles of an n-sided polygon is (n-2)180.
This formula is based on the idea that the interior angles of a triangle sum to 180?.
Given a pentagon:
Draw in the diagonals from one vertex.
3 triangles are formed.
3 x 180 = 540
82. 82 Interior angles of a polygon:
The sum of the measures of the interior angles of an n-sided polygon is (n-2)180.
This formula is based on the idea that the interior angles of a triangle sum to 180?.
Given a pentagon:
Draw in the diagonals from one vertex.
3 triangles are formed.
3 x 180 = 540
Using the formula, we find
(5-2)180 = 3(180) = 540
83. 83 Classification of Triangles:
Triangles are classified by angle measure and side length.
Classification by angle
Acute triangle – a triangle with 3 acute angles.
Obtuse triangle – a triangle with 1 obtuse angle.
Right triangle – a triangle with 1 right angle.
84. 84 Classification of Triangles:
Classification by sides:
Isosceles triangle – a triangle with 2 equal sides (and two equal angles).
Equilateral triangle – a triangle with 3 equal sides (and three equal angles).
Scalene triangle – a triangle where no two sides are equal in length (thus, no two angles are of equal length).
85. 85 Classification of Triangles:
At the top of page 443 in the text is a table illustrating triangles classified by their angles and by their sides. You will be asked to classify triangles in both ways on your homework (page 447 #11-21).
86. 86 Similar Figures:
similar figures – are figures that have the same shape, but may be of different sizes.
Two polygons are similar if their corresponding angles have the same measure and their corresponding sides form a ratio.
87. 87 Similar Figures:
Similar figures can be any polygon, but most of the time, when we talk about similar figures, we are talking about similar triangles.
Study Example 2 on page 443 in the text. It illustrates how to find the missing side of a triangle given a pair of similar triangles. Pay close attention to how the proportions are set up and solved.
88. 88 Example - Similar Figures
89. 89 Example - Similar Figures
90. 90 Example - Similar Figures
91. 91 Example - Similar Figures
92. 92 Example - Similar Figures
93. 93 Example - Similar Figures
94. 94 Example - Similar Figures
95. 95 Example - Similar Figures
96. 96 Congruent Figures:
congruent figures – two similar figures whose corresponding sides are the same length.
When figures are congruent, they are the exact same size, corresponding angles and corresponding sides are congruent.
Read Example 4 on page 445 in the text. It is a example about sides and angles of congruent figures.
97. 97 Quadrilaterals:
quadrilaterals – are four sided polygons.
Since they have four sides, the sum of the interior angles of a quadrilateral is 360? (since (4-2)180 = 360).
Read the chart in the middle of page 446 in the text. It summarizes quadrilaterals and classifies them according to their characteristics. Notice which definitions include parallel sides, congruent sides, and right angles.
98. 98 Quadrilaterals:
After reviewing the chart on page 446 in the text, try to answer the following true/false statements.
99. 99 Quadrilaterals:
After reviewing the chart on page 446 in the text, try to answer the following true/false statements.
1. A trapezoid is a parallelogram.
100. 100 Quadrilaterals:
After reviewing the chart on page 446 in the text, try to answer the following true/false statements.
1. A trapezoid is a parallelogram.
False, a trapezoid only has one pair of opposite sides parallel, parallelograms have both pairs of opposite sides parallel.
101. 101 Quadrilaterals:
After reviewing the chart on page 446 in the text, try to answer the following true/false statements.
2. A rhombus is a parallelogram.
102. 102 Quadrilaterals:
After reviewing the chart on page 446 in the text, try to answer the following true/false statements.
2. A rhombus is a parallelogram.
True, a rhombus has both pairs of opposite sides parallel.
103. 103 Quadrilaterals:
After reviewing the chart on page 446 in the text, try to answer the following true/false statements.
3. A square is a rectangle.
104. 104 Quadrilaterals:
After reviewing the chart on page 446 in the text, try to answer the following true/false statements.
3. A square is a rectangle.
True, a rectangle is a parallelogram whose angles are right angles. A square is also a parallelogram whose angles are right angles.
105. 105 Quadrilaterals:
After reviewing the chart on page 446 in the text, try to answer the following true/false statements.
4. A rectangle is a square.
106. 106 Quadrilaterals:
After reviewing the chart on page 446 in the text, try to answer the following true/false statements.
4. A rectangle is a square.
False, not only do squares have right angles, but the four sides of a square are equal in length. A rectangle does not necessarily have sides of equal length.
107. 107 Quadrilaterals:
After reviewing the chart on page 446 in the text, try to answer the following true/false statements.
5. A rhombus is a square.
False, not only do squares have four sides of equal length, but a square has four right angles. A rhombus does not necessarily have four right angles.
108. 108 Quadrilaterals:
After reviewing the chart on page 446 in the text, try to answer the following true/false statements.
6. A square is a rhombus.
True, a rhombus is a parallelogram with four sides equal in length. A square is also a parallelogram with four sides equal in length.
109. 109 9.3 - Perimeter and Area
Geometric shapes exist in nature and in the world made by human beings. Being able to calculate the perimeter or area of a figure is useful when trying to build a fence, gutter a house, paint a house, even planting a garden. There is an unlimited number of situations where perimeter and area are useful.
110. 110 Perimeter:
The perimeter, P, of a two dimensional figure is the sum of the lengths of the sides of the figure.
Perimeter is the distance around something, and since it is a distance, it is measured in feet, inches, meters, kilometers, etc.
111. 111 Perimeter:
The perimeter, P, of a two dimensional figure is the sum of the lengths of the sides of the figure.
Perimeter is the distance around something, and since it is a distance, it is measured in feet, inches, meters, kilometers, etc.
The perimeter of the trapezoid below would be the sum of the lengths of the 4 sides.
112. 112 Perimeter:
The perimeter, P, of a two dimensional figure is the sum of the lengths of the sides of the figure.
Perimeter is the distance around something, and since it is a distance, it is measured in feet, inches, meters, kilometers, etc.
The perimeter of the trapezoid below would be the sum of the lengths of the 4 sides.
113. 113 Area:
The area, A, is the region within the boundaries of the figure. Area is measured in square units, for example ft2, in2, km2.
114. 114 Area:
The area, A, is the region within the boundaries of the figure. Area is measured in square units, for example ft2, in2, km2.
When calculating area, you want to figure out how many squared units it would take to tile the figure. The green color in the figure below represents what we are calculating when we find the area of the figure.
115. 115 Rectangle Formulas:
Given a rectangle:
Perimeter = 2l + 2w
Area = l x w
116. 116 Square Formula:
Given a square:
Area = s2
117. 117 Parallelogram Formula:
Given a parallelogram:
Area = b x h
118. 118 Triangle Formula:
Given a triangle:
Area = ˝ bh
119. 119 Trapezoid Formula:
Given a trapezoid:
Area = ˝ h(b1 + b2)
120. 120 Perimeter and Area Formulas:
Look at the perimeter and area summary chart at the bottom of page 452 in the text.
It might be useful to write the formulas on a 3x5 card. Use the card for a bookmark for this chapter, and then when you need the formulas, you can find them quickly. There will be more formulas in the next few sections of the text, so write small!
Note: When you take the test on this information, you can use your 3x5 card with the formulas written on it.
121. 121 Perimeter and Area:
1. Find the area of the following figure. Dimensions are in meters.
122. 122 Perimeter and Area:
1. Find the area of the following figure. Dimensions are in meters.
123. 123 Perimeter and Area:
1. Find the area of the following figure. Dimensions are in meters.
124. 124 Perimeter and Area:
1. Find the area of the following figure. Dimensions are in meters.
125. 125 The Pythagorean Theorem:
The Pythagorean theorem is an important tool when finding the areas and perimeters of triangles.
The Pythagorean theorem states: The sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse.
126. 126 The Pythagorean Theorem:
Study Example 2 on page 453 in the text. This problem illustrates how the Pythagorean theorem can be used to find the hypotenuse of a triangle.
The example on the following slide shows how to use the Pythagorean theorem to find the length of one of the legs of a right triangle.
127. 127 The Pythagorean Theorem:
128. 128 The Pythagorean Theorem:
129. 129 The Pythagorean Theorem:
130. 130 Circles:
A circle is a set of points that are equidistant from a fixed point called the center.
A radius, r, of a circle is a line segment from the center of the circle to any point on the circle.
A diameter, d, of a circle is a line segment through the center of a circle with both endpoints on the circle. The length of the diameter is twice the length of the radius.
The circumference is the length of the simple closed curve that forms the circle.
131. 131 Circles:
132. 132 Circles:
133. 133 Circles:
Other circle formulas:
The circumference of a circle, C = 2?r or C = ?d.
134. 134 Circles:
Other circle formulas:
The circumference of a circle, C = 2?r or C = ?d.
The area of a circle, A = ?r2
135. 135 Circles:
2. Find the circumference and area of the circle below.
136. 136 Circles:
2. Find the circumference and area of the circle below.
137. 137 Circles:
2. Find the circumference and area of the circle below.
138. 138 Circles:
There is a nice illustration that summarizes all of the terms for circle in Figure 9.33 on page 454 in the text.
Study Examples 3 & 4 on pages 454 – 456 in the text. These examples illustrate the usefulness of the formulas in this section of the text.
Study Example 5 on page 456 in the text. It is an important example because it illustrates how to convert areas to different units.
139. 139 Other Examples:
3. Find the shaded area in the figure below.
140. 140 Other Examples:
3. Find the shaded area in the figure below.
141. 141 Other Examples:
3. Find the shaded area in the figure below.
142. 142 Other Examples:
3. Find the shaded area in the figure below.
143. 143 Other Examples:
3. Find the shaded area in the figure below.
144. 144 Other Examples:
4. A picture frame 4 in. wide surrounds a portrait that is 11 in. wide by 17 in. high. Find the area of the picture frame.
145. 145 Other Examples:
4. A picture frame 4 in. wide surrounds a portrait that is 11 in. wide by 17 in. high. Find the area of the picture frame.
146. 146 Other Examples:
4. A picture frame 4 in. wide surrounds a portrait that is 11 in. wide by 17 in. high. Find the area of the picture frame.
147. 147 Other Examples:
4. A picture frame 4 in. wide surrounds a portrait that is 11 in. wide by 17 in. high. Find the area of the picture frame.
148. 148 Other Examples:
4. A picture frame 4 in. wide surrounds a portrait that is 11 in. wide by 17 in. high. Find the area of the picture frame.
149. 149 Other Examples:
4. A picture frame 4 in. wide surrounds a portrait that is 11 in. wide by 17 in. high. Find the area of the picture frame.
150. 150 Other Examples:
4. A picture frame 4 in. wide surrounds a portrait that is 11 in. wide by 17 in. high. Find the area of the picture frame.
151. 151 Other Examples:
4. A picture frame 4 in. wide surrounds a portrait that is 11 in. wide by 17 in. high. Find the area of the picture frame.
152. 152 9.4 – Volume
When working with a one-dimensional figure, we can find its length. When working with a two-dimensional figure, we can find its area. This section discusses three-dimensional figures. For a three-dimensional figure (i.e., a rectangular shoebox), we can find its volume.
153. 153 Volume:
Volume is the measure of capacity of a figure. Volumes are measured in cubic units such as cubic inches or cubic meters.
Solid geometry is the study of three-dimensional solid figures. Volume and surface area can be calculated for these three-dimensional figures. This section covers volume.
154. 154 Volume – Rectangular Solid:
Volume of a Rectangular solid:
155. 155 Volume – Cube:
A cube is a rectangular solid with the same length, width, and height.
Volume of a cube:
156. 156 Volume – Cylinder:
Volume of a cylinder:
V=?r2h
157. 157 Volume – Cone:
Volume of a cone:
V=1/3(?r2h)
158. 158 Volume – Cone:
Volume of a cone:
V=1/3(?r2h)
159. 159 Volume – Sphere:
Volume of a sphere:
V=4/3(?r3)
160. 160 Volume Summary:
A summary chart of the preceding five volumes can be found at the bottom of page 461 in the text.
Study Example 1 on page 462 in the text. It shows how to find the volume of a rectangular solid that relates to a real life situation. Notice how to determine how much the concrete would cost.
Study Example 2 on page 462 in the text. It is an interesting problem having to do with cylinders.
161. 161 Polyhedrons:
A polyhedron is a closed surface formed by the union of polygonal regions.
Each polygonal region is called a face of the polyhedron.
The line segment formed by the intersection of two faces is called an edge.
The point at which two or more edges intersect is called a vertex. (think “corners”)
162. 162 Euler’s Polyhedron Formula:
An interesting thing occurs when working with polyhedron.
163. 163 Euler’s Polyhedron Formula:
An interesting thing occurs when working with polyhedron.
164. 164 Euler’s Polyhedron Formula:
An interesting thing occurs when working with polyhedron.
165. 165 Polyhedrons:
A regular polyhedron is a polyhedron whose faces are all regular polygons of the same size and shape (i.e., a cube).
A prism is a special polyhedron whose bases are congruent polygons and whose sides are parallelograms. The parallelogram regions are called the lateral faces of the prism.
A right prism is a prism whose lateral faces are all rectangles.
A few prisms are illustrated in Figure 9.40 page 464 in the text.
166. 166 Volume – Prism:
The volume of any prism can be found by multiplying the area of the base, B, by the height, h, of the prism.
V = Bh
167. 167 Volume – Prism:
The volume of any prism can be found by multiplying the area of the base, B, by the height, h, of the prism.
V = Bh
Study Example 4 on page 464 in the text. This example illustrates how to find the volume of a trapezoidal prism.
168. 168 Volume of Prism – Example:
Find the volume of the prism shown.
169. 169 Volume of Prism – Example:
Find the volume of the prism shown.
170. 170 Volume of Prism – Example:
Find the volume of the prism shown.
171. 171 Volume of Prism – Example:
Find the volume of the prism shown.
172. 172 Volume – Pyramids:
A pyramid is a polyhedron which has only one base. Pyramids are named by their base. If a pyramid has a triangle for its base, it is called a triangular pyramid.
Figure 9.43 on page 465 in the text illustrates four different pyramids. Notice how all of the faces except for one (the base) are triangles.
Figure 9.44 on page 466 in the text shows a pyramid inside of a prism. Notice how the volume of the pyramid is less than the volume of the prism.
173. 173 Volume – Pyramids:
It turns out that it would take three pyramids to equal the volume of the prism with the same base and height.
Volume of a pyramid:
174. 174 Volume – Pyramids:
It turns out that it would take three pyramids to equal the volume of the prism with the same base and height.
Volume of a pyramid:
Study Example 6 on page 466 in the text. This example illustrates how to find the volume of a pyramid with a square base.
175. 175 Volume of Pyramid – Example:
Look at the diagram for problem 18 on page 467 in the text. We are asked to find the volume of this pyramid.
176. 176 Volume of Pyramid – Example:
Look at the diagram for problem 18 on page 467 in the text. We are asked to find the volume of this pyramid.
Remember that the formula for a pyramid is:
V = 1/3(Bh). Since the base is a triangle, the area of the triangular base is B = ˝(bh).
B = ˝(bh) = ˝(9x15) = ˝(135)= 67.5 = B
177. 177 Volume of Pyramid – Example:
Look at the diagram for problem 18 on page 467 in the text. We are asked to find the volume of this pyramid.
Remember that the formula for a pyramid is:
V = 1/3(Bh). Since the base is a triangle, the area of the triangular base is B = ˝(bh).
B = ˝(bh) = ˝(9x15) = ˝(135)= 67.5 = B
V = 1/3(Bh) = 1/3(67.5x13) = 1/3(877.5) = 292.5 ft3
The volume of the pyramid is 292.5 ft3.
178. 178 Volume – Homework Examples:
Look at the diagram for problem 24 on page 468 in the text. We are asked to find the volume of the shaded area.
179. 179 Volume – Homework Examples:
Look at the diagram for problem 24 pm page 468 in the text. We are asked to find the volume of the shaded area.
This is a problem that can be solved using subtraction. Find the volume of the cube, find the volume of the sphere, and then subtract the two answers. To find the volume of the sphere, you need to know the radius of the sphere. Since the sphere is 4 feet across, the radius of the sphere must be 2 feet.
Try to solve this problem. The solution follows.
180. 180 Volume – Homework Examples:
The volume of the cube is V = s3 = 43 = 64 ft3.
181. 181 Volume – Homework Examples:
The volume of the cube is V = s3 = 43 = 64 ft3.
The volume of the cylinder is:
V = 4/3(?r3) = 4/3(3.14 x 23) = 4/3( 3.14 x 8) =
4/3 ( 25.12 ) = 33.49 ft3.
182. 182 Volume – Homework Examples:
The volume of the cube is V = s3 = 43 = 64 ft3.
The volume of the cylinder is:
V = 4/3(?r3) = 4/3(3.14 x 23) = 4/3( 3.14 x 8) =
4/3 ( 25.12 ) = 33.49 ft3.
The volume of the shaded area is found by subtracting the volume of the sphere from the volume of the cube.
183. 183 Volume – Homework Examples:
The volume of the cube is V = s3 = 43 = 64 ft3.
The volume of the sphere is:
V = 4/3(?r3) = 4/3(3.14 x 23) = 4/3( 3.14 x 8) =
4/3 ( 25.12 ) = 33.49 ft3.
Volume of the shaded area is found by subtracting the volume of the sphere from the volume of the cube.
V shaded area = 64 – 33.49 = 30.51 ft3.
184. 184 9.5 - The Möbius Strip, Klein Bottle, and Maps
This section covers a branch of mathematics called topology. Topology is sometimes referred to as “rubber sheet geometry”. Read on and see if you can figure out why.
185. 185 The Möbius Strip:
August Ferdinand Möbius is best known for his studies of the properties of one-sided surfaces, one of which is called a Möbius strip.
A Möbius strip, also called a Möbius band, is a one-sided, one-edged surface.
Follow the instructions on page 471 in the text to make a Möbius strip. Work through the four Experiments on the same page to learn some interesting properties of the Möbius strip.
186. 186 The Möbius Strip:
In Experiment 1, on page 471 in the text, you should have discovered that a Möbius strip has one edge.
In Experiment 2, on page 471 in the text, you should have discovered that a Möbius has one surface.
In Experiment 3, on page 471 in the text, you should have created one long loop (one strip).
In Experiment 4, on page 471 in the text, you should have created a long loop and a shorter loop (two strips).
187. 187 The Klein Bottle:
The punctured Klein bottle resembles a bottle but only has one side. It was named after Felix Klein.
Look closely at the model of the Klein bottle shown in Figure 9.52 on page 472 in the text. The Klein bottle has only one edge and no outside or inside because it has just one side.
Read the second paragraph on page 472 in the text. It starts with, “Imagine trying to paint a Klein bottle…”.
188. 188 Maps:
Maps have interested topologists for years. Map makers have known for a long time that no matter whether a map is drawn on a flat surface or a sphere, and no matter how complex the diagram is, only four colors are needed to differentiate each country (or state) from its immediate neighbor.
189. 189 Maps:
Maps have interested topologists for years. Map makers have known for a long time that no matter whether a map is drawn on a flat surface or a sphere, and no matter how complex the diagram is, only four colors are needed to differentiate each country (or state) from its immediate neighbor.
Meaning, every map can be drawn by using only four colors, and no two countries with a common border will have the same color. Note: countries which meet in a single point are not considered to have a common border.
190. 190 Maps:
Originally no one had proved that you only needed four colors, but it was accepted as being the case as early as 1852. Then, in 1976 two people from the University of Illinois proved the “four-color” problem using a computer.
191. 191 Maps:
Originally no one had proved that you only needed four colors, but it was accepted as being the case as early as 1852. Then, in 1976 two people from the University of Illinois proved the “four-color” problem using a computer.
Read page 473 in the text for a brief description on how they went about solving the “four-color” problem. When you read this, keep in mind what computers were like in 1976, definitely not what we have today.
192. 192 Jordan Curves:
A Jordan curve is a topological object that can be thought of as a circle twisted out of shape. Look at Figure 9.57 on page 474 in the text. This shows the formation of a Jordan curve. Take a circle, deform it, and “spiral” it up.
Since a Jordan curve is a circle, it has an inside and an outside. With a circle, it is easy to tell if a point is inside the circle or outside the circle. But how can you tell if a point is inside or outside of a Jordan curve?
193. 193 Jordan Curves:
A quick way to tell whether a point is inside or outside the curve is to draw a straight line from the point to a point that is clearly outside the curve. If the straight line crosses the curve in an even number of points, the point is outside. If the straight line crosses the curve in an odd number of points, the point is inside the curve.
194. 194 Jordan Curves:
A quick way to tell whether a point is inside or outside the curve is to draw a straight line from the point to a point that is clearly outside the curve. If the straight line crosses the curve in an even number of points, the point is outside. If the straight line crosses the curve in an odd number of points, the point is inside the curve.
Do you know why this works? Can you explain it in words?
195. 195 Jordan Curves:
Here is a Jordan curve. Determine if the point is inside, or outside of the curve.
196. 196 Jordan Curves:
Here is a Jordan curve. Determine if the point is inside, or outside of the curve.
197. 197 Jordan Curves:
Here is a Jordan curve. Determine if the point is inside, or outside of the curve.
198. 198 Jordan Curves:
Here is a Jordan curve. Determine if the point is inside, or outside of the curve.
199. 199 Jordan Curves:
Here is a Jordan curve. Determine if the point is inside, or outside of the curve.
200. 200 Topological Equivalence:
A doughnut equals a coffee cup?
201. 201 Topological Equivalence:
A doughnut equals a coffee cup?
Two geometric figures are said to be topologically equivalent if one figure can be elastically twisted, stretched, bent, or shrunk into the other figure without puncturing or ripping the original figure.
202. 202 Topological Equivalence:
A doughnut equals a coffee cup?
Two geometric figures are said to be topologically equivalent if one figure can be elastically twisted, stretched, bent, or shrunk into the other figure without puncturing or ripping the original figure.
Look at Figure 9.58 on page 474 in the text. This shows how a doughnut is topologically equivalent to a coffee cup. Not something you would normally think about unless you were a topologist!
203. 203 Topological Equivalence:
In topology, figures are classified according to their genus. The genus of an object is determined by the number of holes in the object. If two objects have the same genus, then they are topologically equivalent.
204. 204 Topological Equivalence:
In topology, figures are classified according to their genus. The genus of an object is determined by the number of holes in the object. If two objects have the same genus, then they are topologically equivalent.
So, the doughnut had one hole, and the coffee cup had one hole (the handle), so they were topologically equivalent.
205. 205 Topological Equivalence:
In topology, figures are classified according to their genus. The genus of an object is determined by the number of holes in the object. If two objects have the same genus, then they are topologically equivalent.
So, the doughnut had one hole, and the coffee cup had one hole (the handle), so they were topologically equivalent.
See Figure 9.59 on page 474 in the text to see more objects that are topologically equivalent (Note: read down the columns of the chart).
206. 206 9.6 - Non-Euclidean Geometry and Fractal Geometry
The beginning of this chapter introduced plane geometry. We accepted the postulates and axioms of plane geometry as true. This section discusses what happens when you do not accept these postulates and axioms as true.
207. 207 The Euclidean Parallel Postulate:
Plane geometry is based on Euclid’s fifth postulate.
The fifth postulate stated, “If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.”
A diagram may help clarify what this postulate means.
208. 208 The Euclidean Parallel Postulate:
209. 209 The Euclidean Parallel Postulate:
210. 210 The Euclidean Parallel Postulate:
211. 211 The Euclidean Parallel Postulate:
212. 212 The Euclidean Parallel Postulate:
In 1795, John Playfair, a Scottish physicist and mathematician wrote a geometry book. In this book, he gave a logically equivalent interpretation of Euclid’s fifth postulate. This version is often referred to as Playfair’s postulate or the Euclidean parallel postulate. This postulate is a little easier to understand.
Euclidean parallel postulate: Given a line and a point not on the line, one and only one line can be drawn through the given point parallel to the given line.
213. 213 The Euclidean Parallel Postulate:
Given a line and a point not on the line, one and only one line can be drawn through the given point parallel to the given line.
214. 214 The Euclidean Parallel Postulate:
Given a line and a point not on the line, one and only one line can be drawn through the given point parallel to the given line.
215. 215 The Euclidean Parallel Postulate:
Given a line and a point not on the line, one and only one line can be drawn through the given point parallel to the given line.
216. 216 The Euclidean Parallel Postulate:
Given a line and a point not on the line, one and only one line can be drawn through the given point parallel to the given line.
217. 217 The Euclidean Parallel Postulate:
Given a line and a point not on the line, one and only one line can be drawn through the given point parallel to the given line.
218. 218 The Euclidean Parallel Postulate:
This postulate makes sense to most people today because Euclidean geometry is what is taught in schools.
Many mathematicians that followed Euclid believed that this postulate was not as self-evident as his other nine. Other mathematicians believed that this parallel postulate was not needed at all.
What happens when you do not follow the parallel postulate? You discover non-Euclidean geometry.
219. 219 Non-Euclidean Geometry:
Non-Euclidean geometry makes sense, but it is all about your point of view, and what you accept as true.
Read pages 477 – 480 in the text. This tells about the history of non-Euclidean geometry and shows the models for each of the geometries. Make sure you know the Fifth Axiom of the three different geometries mentioned in the book. Notice that the axioms are worded the same except for the number of parallel lines that can be drawn.
220. 220 Congratulations!
You have now completed the PowerPoint slides for Chapter 9!
