section 11.1 - linear measure

c©Kendra Kilmer June 12, 2009

Section 11.1 - Linear Measure

The English System

Originally, a yard was the distance from the tip of the nose tothe end of an outstetched arm ofan adult person and a foot was the length of a human foot. Sincethen it has gone through manydefinitions until now the definitions are based on the meter.

Unit Equivalent in Other Unitsyard (yd) 3 feetfoot (ft) 12 inchesmile (mi) 1760 yards or 5280 feet

Dimensional Analysis (Unit Analysis)

A process used to convert from one unit of measure to another usingunit ratios (ratios equivalentto 1).

Example 1: Convert each of the following:

a) 200 feet= yards

b) 3.75 yards= inches

c) 8690 feet= miles

d) 940 inches= yards

The Metric System

Unit Symbol Relationship to Base Unitkilometer km 1000 mhectometer km 100 mdekameter dam 10 mmeter m base unitdecimeter dm 0.1 mcentimeter cm 0.01 mmillimeter mm 0.001 m

Approximate Conversions Between English and Metric Systems

• 1 kilometer≈ 0.62 miles

• 1 meter≈ 1.09 yards

• 2.54 centimeters≈ 1 inch

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a) 3.5 km= m

b) 375 cm= hm

c) 765 mm= dm

d) 5.8 km= cm

e) 70 miles/hour= km/hour

f) 100 yards= meters

Example 3: If our money system used metric prefixes and the base unit was adollar, give metric names toeach of the following:

a) dime

b) penny

c) $10 bill

d) $100 bill

e) $1000 bill

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Greatest Possible Error:

Thegreatest possible error (GPE)of a measurement is one-half the smallest unit used.

Example 4: Determine the GPE for each of the following measurements andinterpret.

a) 25 inches

b) 10.8 cm

c) 5.64 m

Distance Properties

1. The distance between any two pointsA andB is greater than or equal to 0, writtenAB ≥ 0.

2. The distance between any two pointsA andB is the same as the distance betweenB andA, writtenAB = BA.

3. For any three pointsA, B, andC, the distance betweenA andB plus the distance betweenB andC isgreater than or equal to the distance betweenA andC, writtenAB+BC ≥ AC.

Distance Around a Plane Figure

The perimeter of a simple closed curve is the length of the curve. If a figure is a polygon, itsperimeter is the sum of the lengths of its sides. A perimeter has linear measure.

Example 5: Find the perimeter of each of the shapes below:

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Example 6: Given a square of any size, stretch a rope tightly around it. Now take the rope off, add 100 inchesto it, and put the extended rope back around the square so thatthe new rope makes a square around theoriginal square. Findd, the distance between the squares.

Circumference of a Circle

A circle is defined as the set of all points in a plane that are the same distance from a given point,thecenter. The perimeter of a circle is itscircumference.

C = 2πr = πd

wherer is the radius,d is the diameter, andπ = 3.14159....

Example 7: Find the circumference of the circle pictured below:

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Arc Length

The length of an arc depends on the radius of the circle and thecentral angle determing the arc.

Example 8: Determine the length of each arc described below of a circle with radiusr:

a) Semi-circle

b) Quarter Circle

c) Arc with central angleθ .

Example 9: Find the following:

a) the radius of a circle that has a circumference of 18π meters.

b) the length of a 35◦ arc of a circle with radius 15 cm.

c) the radius of a circle that has an arc with central angle 85◦ and length of 150 cm.

Section 11.1 Homework Problems: 1, 3, 6, 11, 13, 17, 19, 23, 25-27, 30-32

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Section 11.2 - Areas of Polygons and Circles

Area is measured using square units and the area of a region isthe number of non-overlappingsquare units that covers the region. For instance, a square measuring 1 cm on a side has an area of1 square centimeter denoted 1 cm2.

Areas on a Geoboard

In teaching the concept of area, intuitive activities should preced the development of formulas.

Example 1: Find the area of each of the figures below:

Converting Units of Area

The most commonly used units of area in the English system arethe square inch (in.2), the squarefoot (ft2), the square yard (yd2), and the square mile (mi2). In the metric system, the most com-monly used units are the square millimeter (mm2), the square centimeter (cm2), the square meter(m2), and the square kilometer (km2). We must be careful when coverting between these units.


a) 1 m2 = cm2

b) 9 yd2 = ft2

c) 5 cm2 = mm2

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d) 124 km2 = m2

e) 3000 ft2 = yd2

f) 14,256in2 = yd2

Land Measure

One application of area today is in land measures. The commonunit of land measure in the Englishsystem is the acre. Historically, an acre was the amount of land a man with one horse could plowin one day.

Unit of Area Equivalent in Other Units1 acre 4840 yd2

1 mi2 640 acres

1 a (are) 100 m2

1 ha (hectare) 100a or 10,000m2 or 1hm2

1 km2 1,000,000 m2

Example 3:

a) A square field has a side of 400 yards. Find the area of the field in acres.

b) A square field has a side of 400 meters. Find the area of the field in hectares.

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Area Formulas

Let’s develop the formulas for the areas of specific shapes.

Rectangle

Parallelogram

Triangle

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Trapezoid

Regular Polygon

Circle

Sector of a Circle

r

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Summary

Figure Area Variables

Rectangle A = lw l =length,w =width

Parallelogram A = bh b =base,h =height

Triangle A = 12bh b =base,h =height

Trapezoid A = 12(b1+b2)h b1,b2 =bases,h =height

Regular Polygon A = n(12sa) n =number of sides,a =apothem,s =length of side

Circle A = πr2 r =radius

Sector of a Circle A =θ◦

360◦(πr2) θ =angle;r =radius

Example 4: Find the area of each of the figures below:

Section 11.2 Homework Problems: 4-7, 9, 16, 17, 20-22, 25-35, 37, 40, 42

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Section 11.3 - The Pythagorean Theorem and the Distance Formula

Theorem 11-1 (Pythagorean Theorem)

If a right triangle has legs of lengthsa andb and hypotenuse of lengthc, thenc2 = a2+b2.

Example 1: Proof 1 of the Pythagorean Theorem (hypotenuse outside)

Example 2: Proof 2 of the Pythagorean Theorem (hypotenuse inside)

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Example 3: The size of a rectangular television screen is given as the length of the diagonal of the screen. Ifthe width of the screen is 24 inches, and the height of the screen is 18 inches, what is the length of thediagonal?

Example 4: A pole BD, 28 feet high, is perpendicular to the ground. Two wiresBC andBA, each 35 feetlong, are attahced to the top of the pole and to stakesA andC on the ground. If pointsA, D, andC arecollinear, how far are the stakesA andC from each other?

Special Right Triangles

Property of 45◦−45◦−90◦ triangle: In an isosceles right triangle, if the length of each leg isa,then the hypotenuse has lengtha

√2.

Property of 30◦−60◦−90◦ triangle: In a 30◦−60◦−90◦triangle, the length of the hypotenuse istwo times as long as the leg opposite the 30◦ angle and the leg opposite the 60◦ angle is

√3 times

the shorter leg.

Example 5: Findx andy in the following figures:

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Theorem 11-2 (Converse of the Pythagorean Theorem)If △ABC is a triangle with sides of lengthsa, b,andc such thata2 + b2 = c2, then△ABC is a right triangle with the right angle opposite the side oflengthc.

Example 6: Determine if the following can be the lengths of the sides of aright triangle:

a) 51,68,85

b) 3,4,7

Distance Formula

The distance between the pointsA(x1,y1) andB(x2,y2) is given by

AB =√

(x2− x1)2+(y2− y1)2

Example 7: Let’s convince ourselves of the distance formula.

Example 8: Find the distance between the points(−5,−3) and(2,−1)

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Example 9: Show that(7,4),(−2,1), and(10,−5) are the vertices of an isosceles triangle.

Example 10: Show that(0,6),(−3,0), and(9,−6) are the vertices of a right triangle.

Example 11: Determine whether or not the points(0,5),(1,2), and(2,−1) are collinear.

Section 11.3 Homework Problems: 1, 2, 8, 9, 11, 12, 14-17, 27-30, 36-41, 53-55

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Section 11.4 - Surface Areas

The surface areais the sum of the areas of the faces (lateral and bases) of a three-dimensionalobject. Thelateral surface areais the sum of the areas of the lateral faces.

Surface Area of a Cube

ss

s

The surface area of a cube is , wheres is the length of a side.

Example 1: Find the surface area of a cube with length of a side 8 inches.

Surface Area of a Right Prism with Regularn-gon Bases

h

The surface area of a Right Prism with regularn-gon bases is , wherel is the lengthof a side of the base,h is the height of a lateral face, andB is the area of the base.

Example 2: Find the surface area of a right regular-hexagonal prism with height 7 feet and length of eachside of the hexagon 4 feet.

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Surface Area of a Right Regular Cylinder

r

h

The surface area of a right circular cylinder is , wherer is the radius of the circle,andh is the height of the cylinder.

Example 3: Find the surface area of a right circular cylinder in which the radius of the circular base is 5 cmand the height of the cylinder is 25 cm.

Surface Area of a Right Regular Pyramid

l

b

The surface area of a right regular pyramid is , wheren is the number of sides ofthe regular polygon,b is the length of a side of the base,B is the area of the base, andl is the slantheight.

Example 4: Find the surface area of a right regular triangular pyramid with slant height 5 inches and lengthof side of the base 4 inches.

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Surface Area of a Right Circular Cone

r

hl

The surface area of a right circular cone is , wherer is the radius of the circle, andl is the slant height.

Example 5: Find the surface area of a right circular cone with height 4 cmand radius 3 cm.

Surface Area of a Sphere

r

The surface area of a sphere is ‘ , wherer is the radius of the sphere.

Example 6: Find the surface area of a sphere with diameter 16 inches.

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Example 7: The napkin ring pictured in the following figure is to be resilvered. How many square millime-ters must be covered?

Example 8: The base of a right pyramid is a regular hexagon with sides of length 12 meters. The altitude ofthe pyramid is 9 meters. Find the total surface area of the pyramid.

Section 11.4 Homework Problems: 1-15, 26, 38-40

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Section 11.5 - Volume, Mass, and Temperature

Surface Areais the number of square units covering a three dimensional figure;Volume describeshow much space a three-dimensional figure contains.

The unit of measure for volume must be a shape that tessellates space (can be stacked so that theyleave no gaps and fill space). Standard units of volume are based on cubes and arecubic units. Acubic unit is the amount of space enclosed within a cube that measures 1 unit on a side.

Example 1: Determine the surface area and volume of the following figure:

Volume of Right Rectangular Prisms

The volume of a right rectangular prism can be measured by determining how many cubes areneeded to build it as a solid.

Thus, the volume of aright rectangular prism is wherel is the length of thebase,w is the width of the base, andh is the height of the prism.

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Converting Metric Measures of Volume

The most commonly used metric units of volume are thecubic centimeterand thecubic meter.


a) 5m3 = cm3

b) 12,300cm3 = m3

In the metric system, cubic units may be used for either dry orliquid measure, although units suchas liters and milliliters are usually used for liquid measures. By definition, aliter , symbolized L,equals the capacity of a cubic decimeter (1L= 1dm3)


a) 3cm3 = L

b) 1mL= cm3

c) 4.2kL= m3

d) 68L= mL

e) 9m3 = L

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Converting English Measures of Volume

Basic units of volume in the English system are the cubic foot(1ft3), the cubic yard (1yd3), andthe cubic inch (1in3). For liquid measures we use the gallon and the quart.

1 gallon= 231 in3

1 quart= 14 gallon


a) 45yd3 = ft3

b) 4320in3 = yd3

c) 3ft3 = yd3

d) 1.3ft3 = gallons

d) 30ft3 = quarts

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Volumes of Specific Figures:

Right Prism

h

The volume of a Right Prism is , whereh is the height of the prism andB is the areaof the base.

Example 5: Find the volume of a right regular-hexagonal prism with height 7 feet and length of each side ofthe hexagon 4 feet.

Right Circular Cylinder

r

h

The volume of a right circular cylinder is , wherer is the radius of the circle, andh is the height of the cylinder.

Example 6: Find the volume of a right circular cylinder in which the radius of the circular base is 5 cm andthe height of the cylinder is 25 cm.

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Right Pyramid

l

b

The volume of a right pyramid is , whereh is the height of the pyramid andB isthe area of the base.

Example 7: Find the volume of a right regular triangular pyramid with height 5 inches and length of side ofthe base 4 inches.

Right Circular Cone

r

hl

The volume of a right circular cone is , wherer is the radius of the circle, andh isthe height of the cone.

Example 8: Find the volume of a right circular cone with height 4 cm and radius 3 cm.

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Sphere

r

The volume of a sphere is ‘ , wherer is the radius of the sphere.

Example 9: Find the volume of a sphere with diameter 16 inches.

Mass

Mass is a quantity of matter.Weight is a force exerted by gravitational pull. On Earth, the termsare commonly interchanged.

In the metric system, the fundamental unit for mass is thegram, denoted g. A paper clip and athumbtack each have a mass of about 1 gram. One mL of water weighs about 1 gram.

Example 10: How many liters of water can a 90 cm by 160 cm by 65 cm rectangular prism hold? What isthe mass in kilograms?

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Temperature

To measure temperature in the metric system thedegree Celsiusis used. To measure temperature inthe English System, theFahrenheit scaleis used. These two scales have the following relationship:

Example 11: Find an equation giving the relationship between the Celsius and Fahrenheit scales, in terms ofCelsius temperature. Use it to convert 65◦ C to ◦F.

Example 12: Find an equation giving the relationship between the Celsius and Fahrenheit scales, in terms ofFahrenheit. Use it to convert 100◦F to ◦C.

Section 11.5 Homework Problems: 1-7, 9-16, 20, 22, 24, 25, 34, 35, 43, 44, 46-48, 63-66

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section 11.1 - linear measure

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