perimeter

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9

THE NATURE OF MEASUREMENT

Copyright © Cengage Learning. All rights reserved.

9.1

Perimeter

Copyright © Cengage Learning. All rights reserved.

Measuring Length

3

Measuring Length

The most common system of measurement used in the world is what we call the

metric system.

There have been numerous attempts to make the metric system mandatory in the United States.

4

Measuring Length

The most difficult problem in changing from the

United States system

(the customary system of measurement in the United States) to the metric system is not mathematical, but psychological. Many people fear that changing to the metric system will require complex multiplying and dividing and the use of confusing decimal points.

5

Measuring Length

The real advantage of using the metric system is the ease of conversion from one unit of measurement to another.

The following box gives the standard units of length.

6

Measuring Length

The basic unit of measurement for the U.S. system is the inch. The basic unit of measurement for the metric system is the meter. 7

Measuring Length

Greek prefixes

kilo-, hecto-

, and

deka-

are used for measurements larger than the basic metric unit, and Latin prefixes

deci-, centi-

, and

milli-

are used for smaller quantities (see Figure 9.3).

Metric prefixes.

Figure 9.3

8

Measuring Length

As you can see from Figure 9.3, a centimeter is of a meter; this means that 1 meter is equal to 100 centimeters.

9

Example 1 –

Use a ruler to find a length of segment

Measure the given segment

a.

to the nearest centimeter.

b.

to the nearest centimeter.

Solution:

a.

b.

10

Perimeter

11

Perimeter

One application of both measurement and geometry involves finding the distance around a polygon. This distance is called the

perimeter

of the polygon.

12

Perimeter

Following are some formulas for finding the perimeters of the most common polygons.

An

equilateral triangle

is a triangle with all sides equal in length (see Figure given below)

Equilateral triangle

PERIMETER = 3(SIDE)

P

= 3

s

13

Perimeter

A

rectangle

is a quadrilateral with opposite sides parallel and equal in length (See the Figure given on right side).

A

square

is a quadrilateral with all sides equal in length (See the Figure given on right side).

Rectangle

PERIMETER = 2(LENGTH) + 2(WIDTH)

P

= 2/ + 2

w

Square

PERIMETER = 4(SIDE)

P

= 4

s

14

Example 2 –

Find the perimeter

Find the perimeter of each polygon.

15

Example 2 –

Solution

a.

Rectangle is 2 cm by 9 cm, so

P

= 2

l

+ 2

w

= 2( 9 ) + 2( 2 ) = 18 + 4 = 22 cm 16

Example 2 –

Solution

b.

Rectangle is 2 ft by 4 ft, so

P

= 2

l

+ 2

w

= 2( 4 ) + 2( 2 ) = 8 + 4 = 12 ft

c.

Square, so

P

= 4

s

= 4( 5 ) = 20 mi cont’d 17

Example 2 –

Solution

d.

Equilateral triangle, so

P

= 3

s

= 3( 10 ) = 30 dm

e.

Triangle (add lengths of sides), so

P

= 3 + 4 + 5 = 12 m cont’d 18

Example 3 –

Find the length of a pen

Suppose you have enough material for 70 ft of fence and want to build a rectangular pen 14 ft wide. What is the length of this pen?

Solution: PERIMETER = 2(LENGTH) + 2(WIDTH) This is the formula for perimeter. Fill in the given information.

70 = 2(LENGTH) + 2( 14 ) 70 = 2

l +

28 Let

l

= LENGTH OF PEN 42 = 2

l

21 =

l

The pen will be 21 ft long.

19

Circumference

20

Circumference

A

circle

is the set of all points in a plane a given distance, called the

radius,

from a given point, called the

center.

Although a circle is not a polygon, sometimes we need to find the distance around a circle. This distance is called the

circumference.

21

Circumference

For

any circle

, if you divide the circumference by the diameter, you will get the

same number

(see Figure 9.5).

Circle

Figure 9.5

22

Circumference

This number is given the name  (

pi

). The number  is an irrational number and is sometimes approximated by 3.14 or We need this number  to state a formula for the circumference

C

:

C

=

d

 where

d

= DIAMETER or

C

= 2 

r

where

r

= RADIUS 23

Circumference

For a circle, the

diameter

is twice the radius. This means that, if you know the radius and want to find the diameter, you simply multiply by 2. If you know the diameter and want to find the radius, divide by 2.

24

Example 4 –

Find distance around a figure

Find the distance around each figure.

25

Example 4 –

Solution

a. C = 4  ; a decimal approximation is

C

 The circumference is 4  4(3.14) = 12.56.

ft, which is about 13 ft.

b. C = 28  ; this is about 88 cm.

c. C = 2  (5) = 10  ; this is about 31 m.

d.

This is half of a circle (called a

semicircle

); thus, the curved part is half of the circumference (

C

= 2   6.28), or about 3.14, which is added to the diameter: 3.14 + 2 = 5.14

The distance around the figure is about 5 dm.

26

Example 4 –

Solution

e.

This is one-eighth of a circle. The curved part is one eighth of the circumference [

C

= 2  (30)  188.5],or about 23.6, which is added here to the radius (on both slides): 23.6 + 30 + 30  84 cm cont’d 27

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