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Measurement

Natural Measures and Standard Units

Systems of weights and measures have been used in many partsof the world since ancient times. People measured lengths andweights long before they had rulers and scales.

Ancient Measures of WeightShells and grains such as wheat or rice were often used as unitsof weight. For example, a small item might be said to weigh 300grains of rice. Large weights were often compared to the loadthat could be carried by a man or a pack animal.

Ancient Measures of LengthPeople used natural measures based on the human body to measure length and distance. Some of these units are shown below.

Standard Units of Length and WeightUsing shells and grains to measure weight is not exact. Even if the shells and grains are of the same type, they vary in sizeand weight.

Using body lengths to measure length is also not exact. Bodymeasures depend upon the person who is doing the measuring.The problem is that different persons have hands and arms ofdifferent lengths.

One way to solve this problem is to make standardunits of length and weight. Most rulers are markedoff with inches and centimeters as standard units.Bath scales are marked off using pounds andkilograms as standard units. The standard unitsnever change and are the same for everyone. If twopeople measure the same object using standardunits, their measurements will be the same oralmost the same.

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Measurement

Millet was raised as agrain crop in ancientChina. The Chinese usedmillet seeds to define aunit of weight called thezhu. One zhu was theweight of 100milletseeds,which isabout �5

10�

ounce.

jointcubit

greatspan

finger stretch 2-fingerwidth

1-fingerwidth

fathom

naturalyard

10 2INCHES

10 2 3 4 5 6CM

centimeter scale

inch scale

The Metric System and the U.S. Customary System

About 200 years ago, a system of weights and measures calledthe metric system was developed. It uses standard units forlength, weight, and temperature. In the metric system:

♦ The meter is the standard unit for length. The symbol for ameter is m. A meter is about the width of a front door.

♦ The gram is the standard unit for weight. The symbol for agram is g. A paper clip weighs about �

12� gram.

♦ The Celsius degree is the standard unit for temperature.The symbol for degrees Celsius is °C. Water freezes at 0°Cand boils at 100°C. Normal room temperature is about 20°C.

Scientists almost always use the metric system for measurement.The metric system is easy to use because it is a base-ten system.Larger and smaller units are defined by multiplying or dividingthe units given above by powers of ten: 10, 100, 1,000, and so on.

The metric system is used in most countries around the world.In the United States, the U.S. customary system is used foreveryday purposes. The U.S. customary system uses standardunits like the inch, foot, yard, mile, ounce, pound, and ton.

Measurement

ExamplesExamples All metric units of length are based on the meter.Each unit is defined by multiplying or dividing themeter by a power of 10.

Check Your UnderstandingCheck Your Understanding

1. Which of these units are in the metric system?foot millimeter pound inch gram meter centimeter yard

2. What does the prefix “milli-” mean?3. 2 grams � ? milligrams

about 1 meter

Units of Length Based on the Meter Prefix Meaning

1 decimeter (dm) � �110� meter deci- �1

10�

1 centimeter (cm) � �1100� meter centi- �1

100�

1 millimeter (mm) � �1,0100� meter milli- �1,0

100�

1 kilometer (km) � 1,000 meters kilo- 1,000

Check your answers on page 439.

Note

The U.S. customarysystem is not based onpowers of 10. This makesit more difficult to use thanthe metric system. Forexample, in order tochange inches to yards,you must know that 36 inches equals 1 yard.

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Converting Units of Length

The table below shows how different units of length in the metricsystem compare. You can use this table to rewrite a length usinga different unit.

The table below shows how different units of length in the U.S.customary system compare. You can use this table to rewrite alength using a different unit.

Measurement

Comparing Metric Units of LengthSymbols for Units

of Length

1 cm � 10 mm 1 m � 1,000 mm 1 m � 100 cm 1 km � 1,000 m mm � millimeter cm � centimeter

1 mm � �110� cm 1 mm � �1,0

100� m 1 cm � �1

100� m 1 m � �1,0

100� km m � meter km � kilometer

Comparing U.S. Customary Units of LengthSymbols for Units

of Length

1 ft � 12 in. 1 yd � 36 in. 1 yd � 3 ft 1 mi � 5,280 ft in. � inch ft � foot

1 in. � �112� ft 1 in. � �3

16� yd 1 ft � �

13� yd 1 ft � �5,2

180� mi yd � yard mi � mile

ExamplesExamples Use the table to rewrite each length using a different unit. Replace theunit given first with an equal length that uses the new unit.

ExamplesExamples Use the table to rewrite each length using a different unit. Replace theunit given first with an equal length that uses the new unit.

Problem Solution

14 ft � ? inches 14 ft � 14 * 12 in. � 168 in.21 ft � ? yards 21 ft � 21 * �

13� yd � �

231� yd � 7 yd

7 miles � ? feet 7 mi � 7 * 5,280 ft � 36,960 ft180 inches � ? yards 180 in. � 180 * �3

16� yd � �

13860

� yd � 5 yd

Problem Solution

56 centimeters � ? millimeters 56 cm � 56 * 10 mm � 560 mm56 centimeters � ? meters 56 cm � 56 * �1

100� m � �1

5060� m � 0.56 m

9.3 kilometers � ? meters 9.3 km � 9.3 * 1,000 m � 9,300 m6.9 meters � ? centimeters 6.9 m � 6.9 * 100 cm � 690 cm

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Personal References for Units of Length

Sometimes it is hard to remember just how long a centimeter ora yard is, or how a kilometer and a mile compare. You may nothave a ruler, yardstick, or tape measure handy. When thishappens, you can estimate lengths by using the lengths ofcommon objects and distances that you know well.

Some examples of personal references for length are given below.A good personal reference is something that you see or use often,so you don’t forget it. A good personal reference also doesn’tchange size. For example, a wooden pencil is not a good personalreference for length because it gets shorter as it is sharpened.

Measurement

The thickness of a dime is about 1 mm.

The diameter of a quarter is about 1 in.

Note

The personal referencesfor 1 meter can also beused for 1 yard. 1 yardequals 36 inches, while 1 meter is about 39.37inches. One meter isoften called a “fat yard,”which means 1 yard plus 1 hand width.

The longest wall in theworld is the Great Wall ofChina. It is 2,150 mileslong, the length of 32,000football fields. It wouldtake about 3,800,000 bigsteps for an adult to walkits length.

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Personal References for Metric Units of Length

About 1 millimeter About 1 centimeter

Thickness of a dime Thickness of a crayonThickness of the point of a thumbtack Width of the head of a thumbtackThickness of the thin edge of a Thickness of a pattern blockpaper match

About 1 meter About 1 kilometer

One big step (for an adult) 1,000 big steps (for an adult)Width of a front door Length of 10 football fields Tip of the nose to tip of the thumb, (including the end zones)with arm extended (for an adult)

Personal References for U.S. Customary Units of Length

About 1 inch About 1 foot

Length of a paper clip A man’s shoe lengthWidth (diameter) of a quarter Length of a license plateWidth of a man’s thumb Length of this book

About 1 yard About 1 mile

One big step (for an adult) 2,000 average-size steps(for an adult)Width of a front doorLength of 15 football fields (including the end zones)Tip of the nose to tip of the thumb,

with arm extended (for an adult)

Perimeter

Sometimes we want to know the distance around a shape,which is called the perimeter of the shape. To measureperimeter, use units of length such as inches, meters, or miles.

To find the perimeter of a polygon, add the lengths of all its sides.Remember to name the unit of length used to measure the shape.

Measurement

ExampleExample Find the perimeter of polygon ABCDE.

2 cm � 2 cm � 1.5 cm � 2 cm � 2.5 cm � 10 cm

The perimeter is 10 centimeters.

Perimeter FormulasRectangles Squares

p = 2 * (l + w) p = 4 * sp is the perimeter, l is the length, p is the perimeter, s is the length w is the width of the rectangle. of one of the sides of the square.

AB

C

D

E

2 cm

2 cm

2.5 cm 2 cm

1.5 cm

ExamplesExamples Find the perimeter of each polygon.

Rectangle Square

Use the formula p � 2 * (l � w). Use the formula p � 4 * s.• length (l ) � 4 cm • length of side (s) � 9 ft

• width (w) � 3 cm • perimeter (p) � 4 * 9 ft

• perimeter (p) � 2 * (4 cm � 3 cm) � 36 ft� 2 * 7 cm � 14 cm

The perimeter is 14 centimeters. The perimeter is 36 feet.

4 cm

3 cm

9 ft


1. Find the perimeter of a rectangle whose dimensions are 4 feet 2 inchesand 9 feet 5 inches.

2. Measure the sides of this book to the nearest half-inch. What is theperimeter of the book?


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1. Measure the diameter of the quarter in millimeters.2. Find the circumference of the quarter in millimeters.3. What is the circumference of a pizza with a 14-inch diameter?

Circumference

The perimeter of a circle is the distance around the circle.

The perimeter of a circle has a special name. It is called thecircumference of the circle.

\

The diameter of a circle is any line segment that passes throughthe center of the circle and has both endpoints on the circle.

The length of a diameter segment is also called the diameter.

If you know the diameter, there is a simple formula for findingthe circumference of a circle.

circumference = pi * diameter or c = � * d(c is the circumference and d is the diameter)

The Greek letter π is called pi. It is approximately equal to 3.14.In working with the number π, you can either use 3.14 or 3�

17� as

approximate values for π, or a calculator with a π key.

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Measurement

ExampleExample Most food cans are cylinders. Their tops andbottoms have circular shapes. The circumferenceof a circular can top is how far the can turnswhen opened by a can opener.

ExampleExample Find the circumference of the circle.

Use the formula c � π * d.• diameter (d) � 8 cm• circumference (c) � π * 8 cm

Use either the π key on the calculator or use 3.14 as an approximate value for π.

Circumference (c) � 25.1 cm, rounded to the nearest tenth of a centimeter.

The circumference of the circle is 25.1 cm.

start

end

diameterci r

c u m f e r e nc

e

8 cm


Area

Area is a measure of the amount of surface inside a closedboundary. You can find the area by counting the number ofsquares of a certain size that cover the region inside theboundary. The squares must cover the entire region. They mustnot overlap, have any gaps, or extend outside the boundary.

Sometimes a region cannot be covered by an exact number ofsquares. In that case, count the number of whole squares andfractions of squares that cover the region.

Area is reported in square units. Units of area for small regionsare square inches (in.2), square feet (ft2), square yards (yd2),square centimeters (cm2), and square meters (m2). For largeregions, square miles (mi2) are used in the United States, whilesquare kilometers (km2) are used in other countries.

You may report area using any of the square units. But youshould choose a square unit that makes sense for the regionbeing measured.

Although each of the measurements above is correct, reportingthe area in square inches really doesn’t give a good idea aboutthe size of the field. It is hard to imagine 7,776,000 of anything!

188 one hundred eighty-eight

Measurement

ExamplesExamples The area of a field-hockey field is reported below in three different ways.

Area of the field is 6,000 square yards.

Area � 6,000 yd2

Area of the field is 54,000 square feet.

Area � 54,000 ft2

Area of the field is7,776,000 square inches.

Area � 7,776,000 in.2

100 yd

60 yd

300 ft

180 ft

3,600 in.

2,160 in.

1 cm

1 cm

1 square centimeter(actual size)

1 in.

1 in.

1 square inch(actual size)

The International SpaceStation (ISS) orbits theEarth at an altitude of250 miles. It is 356 feetwide and 290 feet long,and has an area of over100,000 square feet.


Find the area of these figures. Include the unit in each answer.

1. 2. 3.

Area of a Rectangle

When you cover a rectangular shape with unit squares, thesquares can be arranged into rows. Each row will contain thesame number of squares and fractions of squares.

To find the area of a rectangle, use either of these formulas:

Area � (the number of squares in 1 row) * (the number of rows)Area � length of a base * height

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Measurement

ExampleExample Find the area of the rectangle.

3 rows with 5 squares in each row for a total of 15 squares

Area � 15 square units

5 squares in a row 3 rows

Area Formulas

Rectangles Squares

A = b * h A = s2

A is the area, b is the length of a A is the area, s is the length of a base, h is the height of the rectangle. side of the square.

Either pair of parallel sides in a rectangle can be chosen as its bases.The height of a rectangleis the shortest distancebetween its bases.

ExamplesExamples Find the area of the rectangle. Find the area of the square.

Use the formula A � b * h. Use the formula A � s2.• length of base (b) � 4 in. • length of a side (s) � 9 ft• height (h) � 3 in. • area (A) � 9 ft * 9 ft • area (A) � 4 in. * 3 in. � 81 ft2

� 12 in.2

The area of the rectangle is 12 in.2. The area of the square is 81 ft2.

4 in.

3 in.

9 ft

height

base


3 units

2 units7 m

4 in.7 m

9 in.12

The Rectangle Method of Finding Area

Many times you will need to find the area of a polygon that is nota rectangle. Unit squares will not fit neatly inside the figure, andyou won’t be able to use the formula for the area of a rectangle.

One approach that works well in cases such as these is calledthe rectangle method. Rectangles are used to surround thefigure or parts of the figure. Then the only areas that you mustcalculate are for rectangles and triangular halves of rectangles.

190 one hundred ninety

Measurement

ExampleExample What is the area of triangle JKL?

Draw a rectangle around the triangle.Rectangle JKLM surrounds the triangle.

The area of rectangle JKLM is 10 square units.The segment JL divides the rectangle into twocongruent triangles that have the same area.

The area of triangle JKL is 5 square units.

ExampleExample What is the area of triangle ABC?

Step 4: Add the areas of the two shaded parts: 4�12

� � 3 � 7�12

� square units.

The area of triangle ABC is 7�12� square units.

Step 1: Divide triangleABC into two parts.

Step 2: Draw a rectanglearound the left shaded part.The area of the rectangle is 9 square units. The shadedarea is 4�

12

� square units.

Step 3: Draw a rectanglearound the right shaded part.The area of the rectangle is 6 square units. The shadedarea is 3 square units.

B

A C

B

A C

B

A C

K L

J M

B

A C

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Measurement

ExampleExample What is the area of triangle XYZ?

Step 4: Subtract the areas of the two shaded triangles from the area of rectangle XRYS:12 � 6 � 3 � 3 square units.

The area of triangle XYZ is 3 square units.

Step 1: Draw a rectanglearound the triangle.

Step 2: The area ofrectangle XRYS is 12 squareunits. So, the area of triangleXRY is 6 square units.

Step 3: Draw a rectanglearound triangle ZSY. The areaof the rectangle is 6 squareunits, so the area of triangleZSY is 3 square units.


Use the rectangle method to find the area of each figure below.

Y

ZX

R

S

Y

ZX

R

S

Y

X S

Y

Z


1. 2. 3.

Area of a Parallelogram

In a parallelogram, either pair of opposite sides can be chosenas its bases. The height of the parallelogram is the shortestdistance between the two bases.

In the parallelograms at the right, the height is shown by adashed line that is perpendicular (at a right angle) to thebase. In the second parallelogram, the base has been extendedand the dashed height line falls outside the parallelogram.

Any parallelogram can be cut into two pieces that will form arectangle. This rectangle will have the same base length andheight as the parallelogram. Therectangle will also have the same areaas the parallelogram. So you can findthe area of the parallelogram in thesame way you find the area of therectangle—by multiplying the lengthof the base by the height.

Measurement

Formula for the Area of a Parallelogram

A = b * hA is the area, b is the length of the base, h is the height of the parallelogram.

ExampleExample Find the area of the parallelogram.

Use the formula A � b * h.• length of base (b) � 6 cm• height (h) � 2.5 cm• area (A) � 6 cm * 2.5 cm � 15 cm2

The area of the parallelogram is 15 cm2.

height

base

2.5 cm

6 cm

height

base base

height

height

base


Find the area of each parallelogram. Include the unit in each answer.1. 2. 3.

24 ft

32 ft

10 in.

8 in.

3.8 cm 2.2 cm

1 cm


192 one hundred ninety-two


Find the area of each triangle. Include the unit in each answer.

1. 2. 3.

Area of a Triangle

Any of the sides of a triangle can be chosen as its base.The height of the triangle is the shortest distancebetween the chosen base and the vertex opposite thisbase. The height is shown by a dashed line that isperpendicular (at a right angle) to the base. In some triangles, the base is extended and the dashedheight line falls outside the triangle. In the righttriangle shown, the height line is one of the sides ofthe triangle.

Any triangle can be combined with a second triangle of thesame size and shape to form a parallelogram. Each triangleat the right has the same size base and height as theparallelogram. The area of each triangle is half the area ofthe parallelogram. Therefore, the area of a triangle is halfthe product of the base length multiplied by the height.

one hundred ninety-three 193

Measurement

Area Formulas

Parallelograms Triangles

A = b * h A = �1

2� * (b * h)

A is the area, b is the length A is the area, b is the length of a base, h is the height. of the base, h is the height.

ExampleExample Find the area of the triangle.

Use the formula A � �12

� * (b * h).• length of base (b) � 7 in.• height (h) � 4 in.• area (A) � �

12

� * (7 in. * 4 in.) � �12

� * 28 in.2 � �228� in.2 � 14 in.2

The area of the triangle is 14 in.2.


base

height

heig

ht

base

heig

ht

base

7 in.

4 in

.

height

base

6 in.

8 in.

10 in.

6 cm

9 cm

4.5 yd

4.8 yd

Area of a Circle

The radius of a circle is any line segment that connects thecenter of the circle with any point on the circle. The length of aradius segment is also called the radius.

The diameter of a circle is any segment that passes throughthe center of the circle and has both endpoints on the circle. The length of a diameter segment is also called the diameter.

If you know either the radius or the diameter, you can find theother length too. Use the following formulas:

diameter � 2 * radius radius � �12� * diameter

If you know the radius, there is a simple formula for finding thearea of a circle:

Area � pi * (radius squared) or A � � * r2

(A is the area and r is the radius.)

The Greek letter π is called pi, and it is approximately equal to3.14. You can either use 3.14 or 3�

17� as approximate values for π,

or a calculator with a π key.

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Measurement

ExampleExample Find the area of the circle.

Use the formula A � π * r 2.• radius (r) � 3 in.• area (A) � π * 3 in. * 3 in.

Use either the π key on the calculator or use 3.14 as an approximate value for π.• area (A) � 28.3 in.2, rounded to the

nearest tenth of a square inch.

The area of the circle is 28.3 in.2.


1. Measure the diameter of the dime in millimeters.2. What is the radius of the dime in millimeters?3. Find the area of the dime in square millimeters.

diam

eter

radius

3 in.


Archimedes of Syacuse(287–212 B.C.) was thegreatest mathematicianof the ancient world. He showed that the areaof a circle could beapproximated by theformula A�(3 �

1700�) * r 2.

Note that 3 �1700� � 3.1428...

is a remarkably goodestimate of π, whichequals 3.1415...

Metric Units

1 liter (L) � 1,000 milliliters (mL)

1 milliliter � �1,0100� liter

1 liter � 1,000 cubic centimeters

1 milliliter � 1 cubic centimeter

U.S. Customary Units

1 gallon (gal) � 4 quarts (qt)1 gallon � 2 half-gallons1 half-gallon � 2 quarts1 quart � 2 pints (pt)1 pint � 2 cups (c)1 cup � 8 fluid ounces (fl oz)1 pint � 16 fluid ounces1 quart � 32 fluid ounces1 half-gallon � 64 fluid ounces1 gallon � 128 fluid ounces

Volume and Capacity

VolumeThe volume of a solid object such as a brick or a ball is ameasure of how much space the object takes up. The volume of a container such as a freezer is a measure of how much the container will hold.

Volume is measured in cubic units, such as cubic inches (in.3), cubic feet (ft3), and cubic centimeters (cm3). It is easy to find the volume of an object that is shaped like a cube orother rectangular prism. For example, picture a container in the shape of a 10-centimeter cube (that is, a cube that is 10 cm by 10 cm by 10 cm). It can be filled with exactly 1,000centimeter cubes. Therefore, the volume of a 10-centimeter cube is 1,000 cubic centimeters (1,000 cm3).

To find the volume of a rectangular prism, all you need toknow are the length and width of its base and its height. The length, width, and height are called the dimensionsof the prism.

You can also find the volume of another solid, such as atriangular prism, pyramid, cone, or sphere, by measuring its dimensions. It is even possible to find the volume of an irregular object such as a rock or your own body.

CapacityWe often measure things that can be poured into or out of containers such as liquids, grains, salt, and soon. The volume of a container that is filled with aliquid or a solid that can be poured is often called itscapacity.

Capacity is usually measured in units such asgallons, quarts, pints, cups, fluid ounces, liters,and milliliters.

The tables at the right compare different units ofcapacity. These units of capacity are not cubic units,but liters and milliliters are easily converted to cubic units:

1 milliter � 1 cm3 1 liter � 1,000 cm3

Measurement

The dimensions of arectangular prism

one hundred ninety-five 195

length width

height

1 cm3 1,000 cm3

Volume of a Geometric Solid

You can think of the volume of a geometric solid as the totalnumber of whole unit cubes and fractions of unit cubes neededto fill the interior of the solid without any gaps or overlaps.

Prisms and CylindersIn a prism or cylinder, the cubes can be arranged in layers thateach contain the same number of cubes or fractions of cubes.

The height of a prism or cylinder is the shortest distancebetween its bases. The volume of a prism or cylinder is theproduct of the area of the base (the number of cubes in onelayer) multiplied by its height (the number of layers).

Pyramids and ConesThe height of a pyramid or cone is the shortest distancebetween its base and the vertex opposite its base.

If a prism and a pyramid have the same size base and height,then the volume of the pyramid is one-third the volume of theprism. If a cylinder and a cone have the same size base andheight, then the volume of the cone is one-third the volume of the cylinder.

196 one hundred ninety-six

Measurement

ExampleExample Find the volume of the prism.

3 layers with 8 cubes in each layer makes a total of 24 cubes.

Volume � 24 cubic units

8 cubes in 1 layer 3 layers

base base

height height

heig

htbase

heig

ht

base

base

base

sam

e he

ight

same base area

sam

e he

ight

same base area

The area of the PacificOcean is about 64 millionsquare miles. Theaverage depth of thatocean is about 2.5 miles.So, the volume of thePacific Ocean is about 64 million mi2 * 2.5 mi,or 160 million cubicmiles.

ExampleExample Find the volume of the triangular prism.

Step 1: Find the area of the base (B). Use the formula A � �12

� (b * h).• length of the triangular base (b) � 5 in.• height of the triangular base (h) � 4 in.• area of base (B) � �

12

� * (5 in. * 4 in.) � 10 in.2

Step 2: Multiply the area of the base by the height of the triangular prism. Use the formula V � B * h.• area of base (B) � 10 in.2

• height of prism (h) � 6 in.• volume (V) � 10 in.2 * 6 in. � 60 in.3

The volume of the triangular prism is 60 in.3.

5 in.6 in.

4 in.


Find the volume of each prism. Include the unit in each answer.

1. 2. 3.

2 yd

7 yd

3 yd10 cm

10 cm

10 cm


Volume of a Rectangular or Triangular Prism

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Measurement

Volume of a Prism Area of a Rectangle Area of a Triangle

V = B * h A = b * h A = �1

2� * (b * h)

V is the volume, B is the area A is the area, b is the length A is the area, b is the length of the base, h is the height of of the base, h is the height of of the base, h is the height of the prism. the rectangle. the triangle.

ExampleExample Find the volume of the rectangular prism.

Step 1: Find the area of the base (B). Use the formula A � b * h.• length of the rectangular base (b) � 8 cm• height of the rectangular base (h) � 5 cm• area of base (B) � 8 cm * 5 cm � 40 cm2

Step 2: Multiply the area of the base by the height of the rectangular prism. Use the formula V � B * h.• area of base (B) � 40 cm2

• height of prism (h) � 6 cm• volume (V) � 40 cm2 * 6 cm � 240 cm3

The volume of the rectangular prism is 240 cm3.

6 cm

5 cm8 cm

8 ft

6 ft12

ft

Volume of a Cylinder or Cone

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Measurement

Volume of a Cylinder Volume of a Cone Area of a Circle

V = B * h V = �1

3� * (B * h) A = π * r 2

V is the volume, B is the area V is the volume, B is the area A is the area, r is the radius of of the base, h is the height of of the base, h is the height of the circle.the cylinder. the cone.

ExampleExample Find the volume of the cylinder.

Step 1: Find the area of the base (B ). Use the formula A � π * r2.• radius of base (r) � 5 cm• area of base (B ) � π * 5 cm * 5 cm

Use either the π key on a calculator or 3.14 as an approximate value for π.• area of base (B ) � 78.5 cm2, rounded to the nearest tenth

of a square centimeter.

Step 2: Multiply the area of the base by the height of the cylinder. Use the formula V � B * h.• area of base (B ) � 78.5 cm2

• height of cylinder (h) � 4 cm• volume (V) � 78.5 cm2 * 4 cm � 314.0 cm3

The volume of the cylinder is 314.0 cm3.

ExampleExample Find the volume of the cone.

Step 1: Find the area of the base (B ). Use the formula A � π * r2.• radius of base (r) � 3 in.• area of base (B ) � π * 3 in. * 3 in.

Use either the π key on a calculator or 3.14 as an approximate value for π.• area of base (B ) � 28.3 in.2, rounded to the nearest tenth of

a square inch.

Step 2: Find �13

� of the product of the area of the base multiplied by the height of the cone.

Use the formula V � �13

� * (B * h).• area of base (B ) � 28.3 in.2

• height of cone (h) � 6 in.• volume (V) � �

13

� * (28.3 in.2 * 6 in.) � 56.6 in.3

The volume of the cone is 56.6 in.3.

4 cm

5 cm

6 in.3 in.

ExampleExample Find the volume of the triangular pyramid.

Step 1: Find the area of the base (B ). Use the formula A � �12

� * (b * h)• length of base (b) � 10 in.• height of base (h) � 6 in.• area of base (B ) � �

12

� * (10 in. * 6 in.) � 30 in.2

Step 2: Find �13

� of the product of the area of the base multiplied by the height of the pyramid. Use the formula V � �

13

� * (B * h).• area of base (B ) � 30 in.2

• height of pyramid (h) � 4�12

� in.• volume (V) � �

13

� * (30 in.2 * 4�12

� in.) � 45 in.3

The volume of the triangular pyramid is 45 in.3.

10 in.

6 in. 4 in.12


Find the volume of each pyramid. Include the unit in each answer.1. 2. 3.


4 yd

Area of base = 96 yd2 4 cm5 cm

12 cm

15 ft

6 ft5 ft

one hundred ninety-nine 199

Volume of a Rectangular or Triangular Pyramid

Measurement

Volume of a Pyramid Area of a Rectangle Area of a Triangle

V = �1

3� * (B * h) A = b * h A = �

1

2� * (b * h)

V is the volume, B is the area A is the area, b is the length A is the area, b is the length of the base, h is the height of of the base, h is the height of of the base, h is the height the pyramid. the rectangle. of the triangle.

ExampleExample Find the volume of the rectangular pyramid.

Step 1: Find the area of the base (B ). Use the formula A � b * h.• length of base (b) � 4 cm• height of base (h) � 2.5 cm• area of base (B ) � 4 cm * 2.5 cm � 10 cm2

Step 2: Find �13

� of the product of the area of the base multiplied by the height of the pyramid. Use the formula V � �

13

� * (B * h).• area of base (B ) � 10 cm2

• height of pyramid (h) � 9 cm• volume (V ) � �

13

� * (10 cm2 * 9 cm) � 30 cm3

The volume of the rectangular pyramid is 30 cm3.

2.5 cm4 cm

9 cm

ExampleExample Find the surface area of the box-like rectangular prism.

Use the formula S � 2 * ((l * w) � (l * h) � (w * h)).• length (l ) � 4 in. width (w) � 3 in. height (h) � 2 in.• surface area (S) � 2 * ((4 in. * 3 in.) � (4 in. * 2 in.) � (3 in. * 2 in.))

� 2 * (12 in.2 � 8 in.2 � 6 in.2)� 2 * 26 in.2 � 52 in.2

The surface area of the rectangular prism is 52 in.2.

3 in.

4 in.

2 in.


Find the surface area of the box-like prism. Include the unit in your answer.

8 cm

10 cm

5 cm


Surface Area of a Rectangular Prism

A rectangular prism has six flat surfaces called faces. Thesurface area of a rectangular prism is the sum of the areas ofall six of its faces. Think of the six faces as three pairs ofopposite, parallel faces. Since opposite faces have the samearea, you find the surface area of one face in each pair ofopposite faces. Then find the sum of these three areas anddouble the result.The simplest rectangular prisms have all six of their facesshaped like rectangles. These prisms look like boxes. You canfind the surface area of a box-like prism if you know itsdimensions: length (l), width (w), and height (h).

Step 1: Find the area of one face in each pair of opposite faces.

Step 2: Find the sum of the areas of the three faces.sum of areas � (l * w) � (l * h) � (w * h)

Step 3: Multiply the sum of the three areas by 2.surface area of prism � 2 * ((l * w) � (l * h) � (w * h))

Measurement

area ofbase � l * w

area of front face � l * h

area of side face � w * h

A box-like prism hasall 6 faces shaped likerectangles.

Surface Area of a Box-Like Rectangular Prism

S = 2 * ((l * w) + (l * h) + (w * h))

S is the surface area, l the length of the base, w the width of the base, h the height of the prism.

h

wl

200 two hundred

Surface Area of a Cylinder

The simplest cylinders look like food cans and are called rightcylinders. Their bases are perpendicular to the line joining thecenters of the bases.

To find the area of the curved surface of a right cylinder, imagine a soup can with a label. If you cut the label perpendicular to the top and bottom of the can, peel it off, and lay it flat on asurface, you will get a rectangle. The length ofthe rectangle is the same as the circumferenceof a base of the cylinder. The width of therectangle is the same as the height of the can.Therefore, the area of the curved surface is theproduct of the circumference of the base andthe height of the can.

The surface area of a cylinder is the sum of the areas of the two bases (2 * π * r2) and the curved surface.

Measurement

Surface Area of a Right Cylinder

S = (2 * π * r 2) + ((2 * π * r ) * h)

S is the surface area, r is the radius of the base, h is the height of the cylinder.

ExampleExample Find the surface area of the right cylinder.

Use the formula S � (2 * π * r 2) � ((2 * π * r) * h).• radius of base (r) � 3 cm• height (h) � 5 cm

Use either the π key on a calculator or 3.14 as an approximate value for π.• surface area (S) � (2 * π * 3 cm * 3 cm) � ((2 * π * 3 cm) * 5 cm)

� (π * 18 cm2) � (π * 30 cm2)� 150.8 cm2, rounded to the nearest tenth of a square centimeter

The surface area of the cylinder is 150.8 cm2.


Find the surface area of the right cylinder to the nearest tenth of a square inch. Include the unit in your answer.

h h

r

c i r c u m f e r e n c ecircumference

circumference of base � 2 * π * rarea of curved surface � (2 * π * r) * h

two hundred one 201


5 cm 3 cm

cylinder

base

base

2 in

.

3 in.

Weight

Today, in the United States, two different sets of standard unitsare used to measure weight.

♦ The standard unit for weight in the metric system is thegram. A small, plastic base-10 cube weighs about 1 gram.Heavier weights are measured in kilograms. One kilogramequals 1,000 grams.

♦ Two standard units for weight in the U.S. customary systemare the ounce and the pound. Heavier weights are measuredin pounds. One pound equals 16 ounces. Some weights arereported in both pounds and ounces. For example, we mightsay that “the suitcase weighs 14 pounds, 6 ounces.”

ExampleExample A bicycle weighs 17 kilograms. How manypounds is that?

Rough Solution: Use the Rule of Thumb. Since 1 kg equalsabout 2 lb, 17 kg equals about 17 * 2 lb � 34 lb.

Exact Solution: Use the exact equivalent. Since 1 kg � 2.205 lb, 17 kg � 17 * 2.205 lb � 37.485 lb.


1. A softball weighs 6 ounces. How manygrams is that? Use both a Rule ofThumb and an exact equivalent.

2. Andy’s brother weighs 58 pounds, 9 ounces. How many ounces is that?

Measurement

Rules of Thumb Exact Equivalents

1 ounce equals about 30 grams. 1 ounce � 28.35 grams1 kilogram equals about 2 pounds. 1 kilogram � 2.205 pounds

Metric Units U.S. Customary Units

1 gram (g) � 1,000 milligrams (mg) 1 pound (lb) � 16 ounces (oz)1 milligram � �1,0

100� gram 1 ounce � �1

16� pound

1 kilogram (kg) � 1,000 grams 1 ton (t) � 2,000 pounds1 gram � �1,0

100� kilogram 1 pound � �2,0

100� ton

1 metric ton (t) � 1,000 kilograms1 kilogram � �1,0

100� metric ton


Note

The Rules of Thumb tableshows how units of weightin the metric system relateto units in the U.S.customary system. Youcan use this table toconvert between ouncesand grams, and betweenpounds and kilograms. Formost everyday purposes,you need only rememberthe simple Rules ofThumb.

lb is an abbreviation for the Latin word libra. Thelibra was the ancient Roman pound and weighed about 0.72 U.S.customary pound.

202 two hundred two

Temperature

Temperature is a measure of the hotness or coldness ofsomething. To read a temperature in degrees, you need areference frame that begins with a zero point and has anumber-line scale. The two most commonly used temperaturescales, Fahrenheit and Celsius, have different zero points.

FahrenheitThis scale was invented in the early 1700s by the Germanphysicist G.D. Fahrenheit. On the Fahrenheit scale, purewater freezes at 32°F and boils at 212°F. A salt-water solutionfreezes at 0°F (the zero point) at sea level. The normaltemperature for the human body is 98.6°F. The Fahrenheitscale is used primarily in the United States.

CelsiusThis scale was developed in 1742 by the Swedish astronomerAnders Celsius. On the Celsius scale, the zero point (0 degreesCelsius or 0°C) is the freezing point of pure water. Pure water boils at 100°C. The Celsius scale divides the intervalbetween these two points into 100 equal parts. For thisreason, it is sometimes called the centigrade scale. The normaltemperature for the human body is 37°C. The Celsius scale is the standard for most people outside of the United Statesand for scientists everywhere.

A thermometer measures temperature. The commonthermometer is a glass tube that contains a liquid. When thetemperature goes up, the liquid expands and moves up thetube. When the temperature goes down, the liquid shrinks and moves down the tube.

Here are two formulas for converting from degrees Fahrenheit (°F) to degrees Celsius (°C) and vice versa:

F � �95� * C � 32 and C � �

59� * (F � 32).

two hundred three 203

Measurement

The thermometers showboth the Fahrenheit andCelsius scales. Key referencetemperatures, such as theboiling and freezing pointsof water, are indicated. A thermometer reading of 70°F (or about 21°C) isnormal room temperature.

ExampleExample Find the Celsius equivalent of 82°F.

Use the formula C � �59

� * (F � 32) and replace F with 82:

C � �59

� * (82 � 32) � �59

� * (50) � 27.77

The Celsius equivalent of 82°F is about 28°C.

220

210

200

190

180

170

160

150

140

130

120

110

100

90

80

70

60

50

40

30

20

10

0

–10

–20

–30

–40

100

90

80

70

60

50

40

30

20

10

0

–10

–20

–30

–40

°F °C

body temperature

roomtemperature

freezing point ofpure water

boiling point ofpure water

freezing pointof saltwater

solution

Measuring and Drawing Angles

Angles are measured in degrees. When writing the measureof an angle, a small raised circle (°) is used as a symbol forthe word degree.

Angles are measured with a tool called a protractor. Youwill find both a full-circle and a half-circle protractor on yourGeometry Template. Since there are 360 degrees in a circle,a 1° angle marks off �3

160� of a circle.

The full-circle protractor on the Geometry Template ismarked off in 5° intervals from 0° to 360°. It can be used tomeasure angles, but it cannot be used to draw angles.

Sometimes you will use a full-circle protractor that is a paper cutout. This can be used to draw angles.

The half-circle protractor on the GeometryTemplate is marked off in 1° intervals from 0° to 180°.

It has two scales, each of which starts at 0°.One scale is read clockwise, the other is read counterclockwise.

The half-circle protractor can be used bothto measure angles and to draw angles.

Two rays starting from the same endpoint form two angles.The smaller angle measures between 0° and 180°. The largerangle is called a reflex angle. It measures between 180° and360°. The sum of the measures of the smaller angle and thereflex angle is 360°.

204 two hundred four

Measurement

100110

120

130

140

150

160

170

180

80 7060

50

4030

2010

0

100 110120

130

140150

160170

180

8070

60

50

40

3020

100

9090

0330

300

270

240

210180

150

12090

60

30

non-reflexangle

reflexangl

e

12 12

3

4567

8

9

1011

Measuring an Angle with a Full-Circle ProtractorThink of the angle as a rotation of the minute hand of a clock. One side of the angle represents the minute hand at the beginning of a time interval. The other side of the angle represents the minute hand some time later.

Measuring an Angle with a Half-Circle Protractor

Measurement

ExampleExample To measure angle ABC with a full-circle protractor:

Step 1: Place the center of the protractor over the vertex of the angle, point B.

Step 2: Line up the 0 mark on the protractor with BA��.

Step 3: Read the degree measure where BC�� crosses the edge of the protractor.

The measure of angle ABC � 30°.

ExampleExample To measure reflex angle EFG:

Step 1: Place the center of the protractor over point F.

Step 2: Line up the 0 mark on the protractor with FG��.

Step 3: Read the degree measure where FE�� crossesthe edge of the protractor.

The measure of angle EFG � 330°.

ExampleExample To measure angle PQR with a half-circle protractor:

Step 1: Lay the baseline of the protractor on QR��.

Step 2: Slide the protractor so that the center of the baseline is over the vertex of the angle, point Q.

Step 3: Read the degree measure whereQP�� crosses the edge of the protractor.There are two scales on the protractor.Use the scale that makes sense for the size of the angle that you are measuring.

90

6030

0

330300

270

240

210

180

150120

A

CB

100110

120

130

140

150

160

170

180

80 7060

50

4030

2010

0

100 110120

130

140150

160170

180

8070

60

50

40

3020

100

9090

P

RQ

9060

300

330

300270

240

210

180

150

120

E

GF

The measure of angle PQR � 50°.

two hundred five 205

ExampleExample Draw a 240° angle.

Step 1: Subtract: 360° � 240° � 120°.

Step 2: Draw a 120° angle.

The larger angle is the reflex angle. It measures 240°.


Measure each angle to the nearest degree.

1. 2. 3.

Draw each angle.

4. 70° angle 5. 280° angle 6. 55° angle

Drawing an Angle with a Half-Circle Protractor

To draw a reflex angle using the half-circle protractor, subtractthe measure of the reflex angle from 360°. Use this as themeasure of the smaller angle.

206 two hundred six

Measurement

ExampleExample Draw a 40° angle.

Step 1: Draw a ray from point A.

Step 2: Lay the baseline of the protractor on the ray.

Step 3: Slide the protractor so that the centerof the baseline is over point A.

Step 4: Make a mark at 40° near theprotractor. There are two scales on theprotractor. Use the scale that makessense for the size of the angle thatyou are drawing.

Step 5: Draw a ray from point Athrough the mark.

100110

120

130

140

150

160

170

180

80 7060

50

4030

2010

0

100 110120

130

140150

160170

180

8070

60

50

40

3020

100

9090

A

mark

120°

240°


ExampleExample What is the sum of the measures of the angles of a hexagon?

Step 1: Draw any hexagon; then divide it into triangles. The hexagon can be divided into four triangles.

Step 2: Multiply the number of triangles by 180°.

4 * 180° � 720°

The sum of the measures of all the angles inside a hexagon equals 720°.


1. Into how many triangles can you dividea. a quadrilateral? b. a pentagon? c. an octagon? d. a 12-sided polygon?

2. What is the sum of the measures of the angles of a pentagon?3. What is the measure of an angle of a regular octagon?4. Suppose that you know the number of sides of a polygon. Without drawing a picture,

how can you calculate the number of triangles into which it can be divided?

The Measures of the Angles of Polygons

Any polygon can be divided into triangles.

♦ The measures of the three angles of each triangle add up to 180°.

♦ To find the sum of the measures of all the angles inside a polygon, multiply: (the number of triangles inside the polygon) * 180°.

Finding the Measure of an Angle of a Regular PolygonAll the angles of a regular polygon have the same measure. Sothe measure of one angle is equal to the sum of the measures ofthe angles of the polygon divided by the number of angles.

two hundred seven 207

Measurement

ExampleExample What is the measure of one angle of a regular hexagon?

The sum of the measures of the angles of any hexagon is 720°. A regular hexagon has 6 congruent angles.

Therefore, the measure of one angle of a regular hexagon is �72

60°� � 120°.

hexagon

regular hexagon (6 congruent sides and 6 congruent angles)



Draw a coordinate grid on graph paper and plot the following points.

1. (2,4) 2. (�1,�3) 3. (0,5) 4. (�2,2)

Plotting Ordered Number Pairs

A rectangular coordinate grid is used to name points in theplane. It is made up of two number lines, called axes, that meetat right angles at their zero points. The point where the two linesmeet is called the origin.

Every point on a rectangular coordinate grid can be named by anordered number pair. The two numbers that make up anordered number pair are called the coordinates of the point. Thefirst coordinate is always the horizontal distance of the point fromthe vertical axis. The second coordinate is always the verticaldistance of the point from the horizontal axis. For example, theordered pair (3,5) names point A on the grid at the right. Thenumbers 3 and 5 are the coordinates of point A.

Measurement

ExampleExample Plot the ordered pair (5,3).

Step 1: Locate 5 on the horizontal axis. Draw a vertical line.

Step 2: Locate 3 on the vertical axis. Draw a horizontal line.

Step 3: The point (5,3) is located at the intersection of the two lines.

The order of the numbers in an ordered pair is important. Thepair (5,3) does not name the same point as the pair (3,5).

ExampleExample Locate (�2,3), (�4,�1), and (3�12�,0).

For each ordered pair:

Locate the first coordinate on the horizontal axis and draw avertical line.

Locate the second coordinate on the vertical axis and draw ahorizontal line.

The two lines intersect at the point named by the ordered pair.

55

4

3

2

1

10 2 3 4 5

A (3,5)

(0,0)

55

4

3

2

1

10 2 3 4 5

A (5,3)

(0,0)


The ordered pair (0,0) names the origin.

55

4

3

2

1

10 2 3 4 5

(3 ,0)12

208 two hundred eight

Latitude and Longitude

The Earth is almost a perfect sphere. All points on Earth areabout the same distance from its center. The Earth rotates onan axis, which is an imaginary line through the center of theEarth connecting the North Pole and the South Pole.

Reference lines are drawn on globes and maps to make placeseasier to find. Lines that go east and west around the Earth arecalled lines of latitude. The equator is a special line oflatitude. Every point on the equator is the same distance fromthe North Pole and the South Pole. The lines of latitude areoften called parallels because each one is a circle that isparallel to the equator.

The latitude of a place is measured in degrees. The symbol fordegrees is (°). Lines north of the equator are labeled °N (degreesnorth), lines south of the equator are labeled °S (degrees south).The number of degrees tells how far north or south of theequator a place is. The area north of the equator is called theNorthern Hemisphere. The area south of the equator is calledthe Southern Hemisphere.

A second set of lines runs from north to south. These aresemicircles (half-circles) that connect the poles. They are calledlines of longitude or meridians. The meridians are notparallel, since they meet at the poles.

The prime meridian is the special meridian labeled 0°. Theprime meridian passes through Greenwich, England (nearLondon). Another special meridian is the international dateline. This meridian is labeled 180° and is exactly opposite theprime meridian on the other side of the world.

two hundred nine 209

Measurement

ExamplesExamples The latitude of the North Pole is 90°N. The latitude of the South Pole is 90°S.

The poles are the points farthest north and farthest south on Earth.

The latitude of Cairo, Egypt, is 30°N. We say that Cairo is 30 degrees north of the equator.

The latitude of Durban, South Africa, is 30°S.Durban is in the Southern Hemisphere.

Parallels(latitude)

Durban

North Pole90°N

South Pole90°S

15°S

30°S

45°S60°S

15°N

30°N

45°N60°N

Equator 0°

Cairo

Northern Hemisphere

Southern Hemisphere

In order to locate placesmore accurately, eachdegree is divided into 60 minutes. One minuteequals �6

10� degree. The

symbol for minutes is ( �).For example, 31°23�Nmeans 31�

2630� degrees north.

ExamplesExamples

This map may be used to find the approximatelatitude and longitude for the cities shown. For example, Denver,Colorado, is about 40°North and 105° West.

210 two hundred ten

Measurement

ExamplesExamples The longitude of Greenwich, England is 0° because it lies on the prime meridian.

The longitude of Durban, South Africa, is30°E. Durban is in the Eastern Hemisphere.

The longitude of Gambia (a small country inAfrica) is about 15°W. We say that Gambiais 15 degrees west of the prime meridian.

The longitude of a place is measured in degrees. Lines west of the prime meridian are labeled °W. Lines east of the primemeridian are labeled °E. The number of degrees tells how far westor east of the prime meridian a place is. The area west of theprime meridian is called the Western Hemisphere. The areaeast of the prime meridian is called the Eastern Hemisphere.

When lines of both latitude and longitude are shown on a globeor map, they form a pattern of crossing lines called a grid. Thegrid can help you locate places on the map. Any place on themap can be located by naming its latitude and longitude.

Meridians(longitude)

GAMBIA

Durban

60°W

45°W

30°W

15°W

15°E

30°E

45°E

60°E

Prim

e M

erid

ian

0°

Ea

st e

r nH

em

isp

he

re

West

ern

He

mi s

ph

er e

Greenwich

70°W90°W 80°W100°W110°W

120°W

120°W 110°W 100°W 90°W 80°W 70°W50°N

40°N

30°N 30°N

40°N

50°N

Atlanta

Denver

Los Angeles

Calgary

Philadelphia

Map Scales and Distances

Map ScalesMapmakers show large areas of land and water on small piecesof paper. Places that are actually thousands of miles apart maybe only inches apart on a map. When you use a map, you canestimate real distances by using a map scale.

Different maps use different scales. On one map, 1 inch mayrepresent 10 miles in the real world. On another map, 1 inchmay represent 100 miles.

On this map scale, the bar is 2 inches long.Two inches on the map represent 2,000 real miles.One inch on the map represents 1,000 real miles.

You may see a map scale written as “2 inches � 2,000 miles.”This statement is not mathematically correct because 2 inches isnot equal to 2,000 miles. What is meant is that a 2-inch distanceon the map represents 2,000 miles in the real world.

Measuring Distances on a MapThere are many ways to measure distances on a map. Here are several.

Use a RulerSometimes the distance you want to measure is along a straightline. Measure the straight-line distance with a ruler. Then usethe map scale to change the map distance to the real distance.

Measurement

ExampleExample Use this map and scale to find the air distance from Denver to Chicago. The air distance is the straight-line distance between the two cities.

The line segment connecting Denver and Chicago is 3 inches long. The map scaleshows that 1 inch represents 300 miles. So 3 inches must represent 3 * 300 miles,or 900 miles. The air distance from Denver to Chicago is 900 miles.

0 1,000 2,000

0 1 2

miles

inches

0 300 600

0 1 2

miles

inches

ScaleChicago

Denver

two hundred eleven 211

Use String and a RulerSometimes you may need to find the length of a curved path,such as a road or river. You can use a piece of string, a ruler,and the map scale to find the length.

♦ Lay a piece of string along the path you want to measure.Mark the beginning and ending points on the string.

♦ Straighten out the string. Be careful not to stretch it. Use aruler to measure between the beginning and ending points.

♦ Use the map scale to change the map distance into the realdistance.

Use a CompassSometimes map scales are not in inches or centimeters, so aruler is not much help. In these cases you can use a compass tofind distances. Using a compass can also be easier than using aruler, especially if you are measuring a curved path and you donot have any string.

Step 1: Adjust the compass so that the distancebetween the anchor point and the pencil pointis the same as a distance on the map scale.

Step 2: Imagine a path connecting the starting pointand ending point of the distance you want tomeasure. Put the anchor point of the compassat the starting point. Use the pencil point tomake an arc on the path. Move the anchorpoint to the spot where the arc and the pathmeet. Continue moving the compass along thepath and making arcs until you reach or passthe ending point. Be careful not to change theopening of the compass.

Step 3: Keep track of how many times you swing thecompass. Each swing stands for the distanceon the map scale. To estimate the totaldistance, multiply the number of swings bythe distance each swing stands for.

If you use a compass to measure distance along a curved path,your estimate will be less than the actual distance. The distancealong a straight line between two points is less than the distancealong a curved path between the same two points.

212 two hundred twelve

Measurement

5

0 1,000 2,000 miles

AB

The compass opening is set to represent 1,000 miles.

The real length of the curve is about 3,000 miles.

Perpetual CalendarThe perpetual calendar consists of 14 different one-yearcalendars. It shows all the possible one-year calendars. Thecalendar for a year is determined by which day is January 1.There are 7 calendars for years with 365 days. There areanother 7 calendars for years with 366 days.

Years that have 366 days are called leap years. They occur everyfour years. The extra day is added to February. Years that aredivisible by 4 are leap years, except for years that are multiples of100. Those years (1600, 1700, 1800, 1900, 2000, and so on) areleap years only if they are divisible by 400. The years 1600 and2000 were leap years, but the years 1700, 1800, and 1900 werenot leap years.

Measurement

Calendar number to use forthe years 1899 to 2028

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJANUARY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SSEPTEMBER

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

S M T W T F SFEBRUARY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SOCTOBER

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SNOVEMBER

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SDECEMBER

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


2

1899 . . . . 11900 . . . . 21901 . . . . 31902 . . . . 41903 . . . . 51904 . . . 151905 . . . . 11906 . . . . 21907 . . . . 31908 . . . 111909 . . . . 61910 . . . . 71911 . . . . 11912 . . . . 91913 . . . . 41914 . . . . 51915 . . . . 61916 . . . 141917 . . . . 21918 . . . . 31919 . . . . 41920 . . . 121921 . . . . 71922 . . . . 11923 . . . . 21924 . . . 10

1925 . . . . 51926 . . . . 61927 . . . . 71928 . . . . 81929 . . . . 31930 . . . . 41931 . . . . 51932 . . . 131933 . . . . 11934 . . . . 21935 . . . . 31936 . . . 111937 . . . . 61938 . . . . 71939 . . . . 11940 . . . . 91941 . . . . 41942 . . . . 51943 . . . . 61944 . . . 141945 . . . . 21946 . . . . 31947 . . . . 41948 . . . 121949 . . . . 71950 . . . . 1

1951 . . . . 21952 . . . 101953 . . . . 51954 . . . . 61955 . . . . 71956 . . . . 81957 . . . . 31958 . . . . 41959 . . . . 51960 . . . 131961 . . . . 11962 . . . . 21963 . . . . 31964 . . . 111965 . . . . 61966 . . . . 71967 . . . . 11968 . . . . 91969 . . . . 41970 . . . . 51971 . . . . 61972 . . . 141973 . . . . 21974 . . . . 31975 . . . . 41976 . . . 12

1977 . . . . 71978 . . . . 11979 . . . . 21980 . . . 101981 . . . . 51982 . . . . 61983 . . . . 71984 . . . . 81985 . . . . 31986 . . . . 41987 . . . . 51988 . . . 131989 . . . . 11990 . . . . 21991 . . . . 31992 . . . 111993 . . . . 61994 . . . . 71995 . . . . 11996 . . . . 91997 . . . . 41998 . . . . 51999 . . . . 82000 . . . 142001 . . . . 22002 . . . . 3

2003 . . . . 42004 . . . 122005 . . . . 72006 . . . . 12007 . . . . 22008 . . . 102009 . . . . 52010 . . . . 62011 . . . . 42012 . . . . 82013 . . . . 32014 . . . . 42015 . . . . 52016 . . . 132017 . . . . 12018 . . . . 22019 . . . . 32020 . . . 112021 . . . . 62022 . . . . 72023 . . . . 12024 . . . . 92025 . . . . 42026 . . . . 52027 . . . . 62028 . . . 14

two hundred thirteen 213

In 46 B.C., Julius Caesarcreated a new calendarbased on the movements of the sun. Pope Gregoryrevised the JulianCalendar in 1582 becausethe year was more than11 minutes too long. The result was theGregorian Calendar, whichproduced the leap yeardivisibility rule.

214 two hundred fourteen

Measurement

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


13

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMAY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SJUNE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SMARCH

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SJULY

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

S M T W T F SAPRIL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

S M T W T F SAUGUST

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31


8

measurementlee5thgrade.pbworks.com/f/measurement.pdf · the metric system and the u.s. customary...

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