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Chapter 9: Geometry
9.1 Points, Lines, Planes, and Angles
9.2 Curves, Polygons, and Circles
9.3 Perimeter, Area, and Circumference
9.4 The Geometry of Triangles: Congruence,
Similarity, and the Pythagorean Theorem
9.5
Space Figures, Volume, and Surface Area
9.6
Transformational Geometry
9.7 Non-Euclidean Geometry, Topology, and Networks
9.8 Chaos and Fractal Geometry
9-3-2
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Perimeter, Area, and Circumference
9-3-3
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Perimeter, Area, and Circumference
• Perimeter of a Polygon
• Area of a Polygon
• Circumference of a Circle
• Area of a Circle
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9-3-4

Perimeter of a Polygon
The perimeter of any polygon is the sum of the measures of the line segments that form its sides. Perimeter is measured in linear units .
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9-3-5

Perimeter of a Triangle
The perimeter P of a triangle with sides of lengths a , b , and c is given by the formula
P = a + b + c.
b a c
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9-3-6

Perimeter of a Rectangle
The perimeter P of a rectangle with length l and width w is given by the formula l
P = 2 l + 2 w, or equivalently w
P = 2( l + w ).
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9-3-7

Perimeter of a Square
The perimeter P of a square with all sides of length s is given by the formula
P = 4 s.
s s
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9-3-8

Area of a Polygon
The amount of plane surface covered by a polygon is called its area . Area is measured in square units .
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9-3-9

Area of a Rectangle
The area A of a rectangle with length l and width w is given by the formula
A = lw.
l w
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9-3-10

Example: Rectangle
Find the perimeter and area of the rectangle below.
15 ft.
7 ft.
Solution
Perimeter
P = 2 l + 2 w = 2(15) + 2(7) = 44 ft.
Area
A = lw = 15(7) = 105 ft.
2
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9-3-11

Area of a Square
The area A of a square with all sides of length s is given by the formula
P = s 2 .
s s
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9-3-12

Area of a Parallelogram
The area A of a parallelogram with height h and base b is given by the formula
A = bh.
h b
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9-3-13

Area of a Trapezoid
The area A of a trapezoid with parallel bases b and B and height h is given by the formula b
A

1
2
  
.
h
B
9-3-14
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Example: Area of a Parallelogram
Find the area of the trapezoid below.
7 cm.
5 cm.
Solution
A

1
2
   
1
2

1
2

  

13 cm.
50 cm.
2
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9-3-15

Area of a Triangle
The area A of a triangle with base b and height h is given by the formula

1
A hb
2
.
h b
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9-3-16

Example: Area With Multiple Shapes
Find the area of the shaded region below.
4 in.
Solution 4 in.
Area of square – Area of triangle
2 
1 s bh
2
4
2 
1
2
(4)(4)
  
8 in.
2
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9-3-17

Circumference of a Circle
The distance around a circle is called its circumference .
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9-3-18

Circumference of a Circle
The circumference C of a circle of diameter d is given by the formula.
C
  d , or equivalently
C

2
 r , where r is a radius.
d r
9-3-19
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Area of a Circle
The area A of a circle with radius r is given by the formula.
A
  r
2
.
r
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9-3-20

Example: Circle
Find the area and circumference of a circle with a radius that is 6 inches long (use 3.14 as an approximation for pi).
Solution
Circumference
C

2
 r
  
12
 
37.68 in.
Area
A
  r
2  
(6)
2 
36
 
113.04 in.
2
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9-3-21