measurement - wikispaces to walk its length. ... length of a paper clip a man’s shoe length width...

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 for centuries before there were rulers and scales.

Ancient Measures of WeightShells and grains such as wheat or rice were often used as units of weight. For example, a small item might be said toweigh 300 grains of rice. Large weights were often compared to the load that 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 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 aremarked off using inches and centimeters asstandard units. Bath scales are marked off usingpounds and kilograms as standard units. Standardunits never change and are the same for everyone.If two people measure the same object usingstandard units, their measurements will be the sameor almost the same.

208 two hundred eight

Measurement

Did You Know?Millet was raised as agrain crop in ancientChina. The Chinese usedmillet seeds to define aunit of weight called thezhu. One zhu was theweight of 100 milletseeds, which is about �510� ounce.

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 a meter is m. A meter is about the width of a front door.

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

12� gram.

♦ The Celsius degree, or �C, is the standard unit fortemperature. Water freezes at 0�C and boils at 100�C.Normal room temperature is about 20�C.

Scientists almost always measure with metric units. The metricsystem is easy to use because it is a base-10 system. Larger andsmaller units are defined by multiplying or dividing the unitsnamed above by powers of 10: 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

Two paper clips weigh about 1 gram.

1. Which of these units are in the metric system?ton millimeter pound mile gram kilometer decimeter ounce

2. a. What does the prefix “kilo”- mean? b. 2 grams � ? kilogramsCheck your answers on page 422.

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

The U.S. customarysystem is not based onpowers of 10. This makes it more difficult to use than the metricsystem. For example, inorder to change inches to yards, you must know that 36 inches equals 1 yard.

Units of Length Based on the Meter Prefix Meaning

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

100�

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

100�

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

10�

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

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Converting Units of Length

This table shows how different units of length in the metricsystem compare. You can use this table to rewrite a lengthusing a 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

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

Use the table to rewrite each length using a different unit. Replacethe unit 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 ft

180 inches � ? yards 180 in. � 180 * �316� yd � �

13860

� yd � 5 yd

210 two hundred ten

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

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

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

Personal References for Units of Length

Sometimes it is difficult to remember exactly how long acentimeter or a yard is or how a kilometer and a milecompare. You may not have a ruler, yardstick, or tapemeasure handy. When this happens, you can estimate lengthsby using the lengths of common objects and distances that youknow well.

Some examples of personal references for length are givenbelow. A good personal reference is something that you see or useoften, 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.

two hundred eleven 211

Measurement

Did You Know?The U.S.–Mexican borderis I,950 miles long, thelength of 29,000 footballfields. It would take about3,500,000 big steps for anadult to walk its length.

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.

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)

Solve. Include the unit in each answer.1. Find the perimeter of a rectangle whose dimensions are 9 feet and 3 feet.2. Find the perimeter of a regular hexagon whose sides are 15 yards long.

Perimeter

The distance around a polygon is called its perimeter. To findthe perimeter of any polygon, add the lengths of all its sides.

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Measurement

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 Formulas

Rectangles Squares Regular Polygons

p � 2 * (l � w) p � 4 * s p � n * sp is the perimeter p is the perimeter p is the perimeterl is the length of the rectangle s is the length of one side n is the number of sidesw is the width of the rectangle of the square s is the length of a side

Check your answers on page 422.

Find the perimeter of each polygon.

Rectangle Square Regular OctagonUse the formula p � 2 * (l � w). Use the formula p � 4 * s. Use the formula p � n * s.

• length (l ) � 4 cm • length of side (s) � 9 ft • number of sides (n) � 8

• width (w) � 3 cm • perimeter (p) � 4 * 9 ft • length of side (s) � �14

� in.

• perimeter (p) � 2 * (4 cm � 3 cm) � 36 ft • perimeter (p) � 8 * �14

� in.

� 2 * 7 cm � 14 cm � �84

� in. � 2 in.

The perimeter is The perimeter is The perimeter is14 centimeters. 36 feet. 2 inches.

Find the circumference of the circle.

Use the formula c � π * d.• diameter (d) � 6 cm• circumference (c) � * 6 cm

Use either the key on a calculator, oruse 3.14 as an approximate value for .

circumference (c) � 18.8 cm, rounded to the nearest tenth of a centimeter

The circumference of the circle is 18.8 cm.

1. Measure the diameter of the dollar coin in millimeters.2. Find the circumference of the dollar coin in millimeters.3. What is the circumference of a pizza whose diameter is 14 inches?

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. Thelength of a diameter segment is also called the diameter.

If you know the diameter of a circle, there is a simple formulafor finding its circumference.

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Measurement

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.

Formula for the Circumference of a Circle

circumference � pi * diameter or c � π * dc is the circumference, and d is the diameter of the circle.


The Greek letter is called pi. It isapproximately equal to3.14. In your work withthe number , you can use 3.14 or 3�

17� as

the approximate value for . You can also use a calculator with a key.

Area

Area is a measure of the amount of surface inside a closedboundary. You can find the area by counting the number of squares of a certain size that cover the region inside theboundary. The squares must cover the entire region. They must not overlap, have any gaps, or cover any surface outsidethe boundary.

Sometimes a region cannot be covered by an exact number ofsquares. In that case, first count the number of whole squares,then the fractions 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 largeareas, square miles (mi2) are used in the United States, whilesquare kilometers (km2) are used in most 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 these measurements is correct, reporting thearea in square inches really doesn’t give us a good idea aboutthe size of the field. It is hard to imagine 7,776,000 of anything!

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Measurement

1 square centimeter (actual size)

1 square inch (actual size)

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.

Did You Know?Tropical rain forests, wheremore than half the plantand animal species in theworld live, once coveredmore than 8,000,000square miles of Earth’ssurface. By 2004, becauseof destruction by people,fewer than 3,400,000square miles remain.

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

Measurement

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.

Find the area of the rectangle.

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

Area � 15 square units

Either pair of parallel sidesin a rectangle can bechosen as its bases. Theheight of a rectangle is theshortest distance betweenits bases.

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.

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Find the area of each parallelogram. Include the unit in each answer.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 extended,and the dashed height line falls outside the parallelogram.

Any parallelogram can be cut into two pieces and the piecesrearranged to form a rectangle whose base length and heightare the same as the base length andheight of the parallelogram. Therectangle has the same area as theparallelogram. So, you can find thearea of the parallelogram in the same way you find the area of therectangle—by multiplying the lengthof the base by the height.

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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.

Find the area of the parallelogram.

Use the formula A � b * h.• length of base (b) � 6 cm• height (h) � 3.8 cm• area (A) � 6 cm * 3.8 cm � 22.8 cm2

The area of the parallelogram is 22.8 cm2.


Area of a Triangle

Any of the sides of a triangle can be chosen as itsbase. The height of the triangle (for that base) isthe shortest distance between the base and thevertex opposite the base.

In the triangles at the right, the height is shown by adashed line that is perpendicular (at a right angle)to the base. In one of the triangles, the base has beenextended and the dashed height line falls outside thetriangle. In the right triangle shown, the height lineis one of the sides of the 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.

Measurement

Area Formulas

Parallelograms Triangles

A � b * h A � �12� * (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.

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.

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

1. 2. 3.


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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 of a circle, youcan find the other length using these formulas:

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

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

Measurement

Formula for the Area of a Circle

Area = pi * (radius squared) or A = π * r 2

A is the area, and r is the radius of the circle.

Find the area of the circle.

Use the formula A � * r 2.• radius (r) � 5 in.• area (A) � * 5 in. * 5 in.

Use either the key on a calculator, oruse 3.14 as an approximate value for .• area (A) � 78.5 in.2, rounded to the

nearest tenth of a square inch.

The area of the circle is 78.5 in.2.

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


218 two hundred eighteen

If d is the diameter, r isthe radius, and c is thecircumference of a circle,then c � * d. Since d � 2 * r, another formulafor circumference is c � 2 * * r.

The Greek letter iscalled pi, and it isapproximately equal to3.14. When you workwith the number , use3.14 or 3�

17� as the

approximate value for ,or use a calculator with a key.

Volume and Capacity

VolumeThe volume of a solid object such as a brick or a ball is a measure of how much space the object takes up. Thevolume of a container such as a freezer is a measure ofhow 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 iseasy to find the volume of an object that is shaped like acube or other rectangular prism. For example, picture acontainer in the shape of a 10-centimeter cube (that is, acube that is 10 cm by 10 cm by 10 cm). It can be filled withexactly 1,000 centimeter cubes. Therefore, the volume of a10-centimeter cube is 1,000 cubic centimeters (1,000 cm3).

All you need to know to find the volume of a rectangularprism 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 other solids (such as triangular prisms, pyramids, cones, and spheres) by measuring their dimensions. It is even possible to find thevolume of irregular objects, such as rocks or your own body.

CapacityWe often measure things that can be poured into or out ofcontainers, such as liquids, grains, salt, and so on. Thevolume of a container that is filled with a liquid or a solidthat can be poured is often called its capacity.

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

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

1 milliliter � 1 cm3 1 liter � 1,000 cm3

Measurement

The Dimensions of a Rectangular Prism

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Metric Units

1 liter (L) � 1,000 (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 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 that areneeded to fill the interior of the solid without gaps or overlaps.

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

The height of a prism or a cylinder is the shortest distancebetween its bases. The volume of a prism or a 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 a 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 ofthe cylinder.

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Measurement

Find the volume of the prism.

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

Volume � 24 cubic units

Did You Know?The area of the AtlanticOcean is about 33 millionsquare miles. Theaverage depth of thatocean is about 2.2 miles.So, the volume of theAtlantic Ocean is about33 million mi2 * 2.2 mi,or 73 million cubic miles.

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.

[EM2007SRB_G6_MEA_T_0009]



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

1. 2. 3.

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 � �12� * (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.

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.


Volume of a Cylinder or Cone

Measurement

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

V � B * h V � �13� * (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.

Find the volume of the cylinder.

Step 1: Find the area of the base (B ). Use the formula A � * r2.

• radius of base (r) � 5 cm• area of base (B ) � * 5 cm * 5 cm

Use the key on a calculator or 3.14 as an approximate value for .

• 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.

Find the volume of the cone.

Step 1: Find the area of the base (B ). Use the formula A � * r2.

• radius of base (r) � 3 in.• area of base (B) � * 3 in. * 3 in.


• 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 ofthe 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.

222 two hundred twenty-two

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 the triangular base (b) � 10 in.• height of the triangular 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.

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

Volume of a Rectangular or Triangular Pyramid

Measurement

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

V � �13� * (B * h) A � b * h A � �

12� * (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.

Find the volume of the rectangular pyramid.

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


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Volume of a Sphere

Suppose you drew point A on a sheet of paper. Now imagine thatyou could draw every point on the sheet of paper that is 2 inchesfrom point A. You would get a circle whose center is point Aand whose radius is 2 inches long. Any points inside the circleare not a part of the circle; they form the interior of the circle.

Now imagine all of the points in space that are 2 inches frompoint A in every direction. You would get a figure that looks likethe surface of a ball. This figure is called a sphere. Point A isthe center of the sphere. The distance from point A to anypoint on the sphere is the radius of the sphere.

Just as with the circle, the points inside the sphere are nota part of the sphere. A good way to think of a sphere is topicture a soap bubble. Another way is to imagine a circlewith a rod passing through its center. If the circle is rotatedaround the rod, the path of the circle will form a sphere.

If a sphere is cut in half, each half is a figure calleda half-sphere. The rim of the half-sphere is a circlewhose center is the center of the sphere.

Measurement

Formula for the Volume of a Sphere

V = �43� * π * r 3

V is the volume and r is the radius of the sphere.

Find the volume of the sphere.

Use the formula V � �43

� * * r3.• radius (r) � 5 cm• volume (V ) � �

43

� * * 5 cm * 5 cm * 5 cm

Use the key on a calculator or 3.14 as an approximate value for .• volume � 523.6 cm3, rounded to the nearest tenth of a cubic centimeter

The volume of the sphere is 523.6 cm3.

Find the volume of each sphere to the nearest tenth of a cubic unit. Include the unit in each answer.1. radius of sphere � 2 inches 2. diameter of sphere � 8 centimeters

224 two hundred twenty-four


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.

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

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 of all six of its faces. Think of the six faces as three pairs ofopposite, parallel faces. Since opposite faces have the samearea, you find the area of one face in each pair of opposite faces.Then find the sum of these three areas and double 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.area of area of front area of sidebase � l * w face � l * h face � w * h

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))

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Measurement

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.

A box-like prism has all 6 faces shaped like rectangles.

Check your answer on page 422.

Find the surface area of the right cylinder.

Use the formula S � (2 * * r 2) � ((2 * * r) * h).

• radius of base (r) � 3 cm• height (h) � 5 cm


• surface area (S) � (2 * * 3 cm * 3 cm) � ((2 * * 3 cm) * 5 cm)� ( * 18 cm2) � ( * 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 nearesttenth of a square inch. Include the unit in your answer.

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 labelperpendicular to the top and bottom of the can, peel it off, andlay it flat on a surface, you will have arectangle. The length of the rectangle is thesame as the circumference of a base of thecylinder. The width of the rectangle is thesame as the height of the can. Therefore, thearea of the curved surface is the product ofthe circumference of the base and the 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.

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.

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

Measurement

Check your answer on page 422.

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right cylinder

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 saltwater 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°C) is thefreezing point of pure water. Pure water boils at 100°C. TheCelsius scale divides the interval between these two points into100 equal parts. For this reason, it is sometimes called thecentigrade scale. The normal temperature for the human bodyis 37°C. The Celsius scale is the standard for most peopleoutside of the United States and 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 shrinksand moves down the tube.

To convert between degrees Fahrenheit (°F) and degreesCelsius (°C), use these formulas:

F � �95� * C � 32 and C � �

59� * (F � 32)

Measurement

These thermometers show both the Fahrenheit and Celsius scales. Keyreference temperatures, such as the boiling and freezing points of water, are indicated. A thermometer reading of 70ºF (or about 21ºC) is normal room temperature.

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.

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Lava that erupts throughthe crater of a volcanousually has a temperaturebetween 1,300°F and 2,300°F.

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.”

Measurement

Solve each problem. Include the unit in each answer.

1. A softball weighs 6 ounces. Howmany grams is that? Use both a Ruleof Thumb and an exact equivalent.

2. Ashley’s sister weighs 36 pounds, 12 ounces. How many ounces is that?


A bicycle weighs 13 kilograms. How many pounds is that?

Rough Solution: Use the Rule of Thumb. Since 1 kg equalsabout 2 lb, 13 kg equals about 13 * 2 lb � 26 lb.

Exact Solution: Use the exact equivalent. Since 1 kg � 2.205 lb, 13 kg � 13 * 2.205 lb � 28.665 lb.

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 � �1,0100� gram 1 ounce � �1

16� pound

1 kilogram (kg) � 1,000 grams 1 ton (t) � 2,000 pounds

1 gram � �1,0100� kilogram 1 pound � �2,0

100� ton

1 metric ton (t) � 1,000 kilograms

1 kilogram � �1,0100� metric ton

The Rules of Thumb Tableshows how units ofweight in the metricsystem compare to unitsin the U.S. customarysystem. You can use thistable to convert betweenounces and grams, andbetween pounds andkilograms. You need onlyremember the simpleRules of Thumb for most everyday purposes.

Did You Know?The EnvironmentalProtection Agencyestimates that about 2 million tons of old andbroken computers, TVs,cell phones, and otherelectronic trash aredumped into U.S.landfills each year. That’smore than 13 pounds perperson each year.

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Capacity and Precision

Different kinds of scales are shown here. The capacity andprecision are shown for each type of scale.

The capacity of a scale is the greatest weight that the scalecan hold. For example, most infant scales have a capacity ofabout 25 pounds. Bath scales are used to weigh older childrenand adults and usually have a capacity of about 300 pounds.

The precision of a scale is its accuracy. If you can read aweight on a bath scale to the nearest pound, then the precisionfor that scale is 1 pound. On most infant scales, you can read aweight to the nearest ounce, so the precision is 1 ounce.

With a balance scale, you can measure weight to the nearestgram. A balance scale is much more precise than an infant scalebecause a gram is lighter than an ounce.

Some scales are extremely precise. They can weigh things thatcannot be seen with the naked eye. Other scales are very large.They can be used to weigh objects that weigh as much as 1,000 tons (2,000,000 pounds). Many scales display weight inboth metric and U.S. customary units.

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Measurement

infant scalecapacity: 25 lbprecision: 1 oz

bath scalecapacity: 300 lb or 135 kgprecision: 0.1 lb or 50 g

balance scalecapacity: 2 kgprecision: 1 g

weight set for balance scale

platform scalecapacity: 1 T to 1,000 T

precision: �14

� lb to 1 T

food scalecapacity: 12 lbprecision: 1oz

spring scalecapacity: 500 gprecision: 20 g

market scalecapacity: 30 lb or 15 kgprecision: 0.01 lb or 5 g

Measuring and Drawing Angles

Angles are measured in degrees. When writing the measure ofan angle, a small raised circle (°) is used as a symbol for theword degree.

Angles are measured with a tool called a protractor. You willfind 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°. Although it can be used to measure angles, it cannot be used to draw angles of a given measure.

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

The half-circle protractor on the GeometryTemplate is marked off in 1° intervals from 0° to180°.

It has two scales. Each scale starts at 0°. Onescale is read clockwise, the other is readcounterclockwise.

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

Two rays starting from the same endpoint form two angles.The smaller angle measures between 0° and 180°. The largerangle measures between 180° and 360°. The larger angle iscalled a reflex angle. The sum of the measures of thesmaller angle and the reflex angle is 360°.

230 two hundred thirty

Measurement

Measuring an Angle with a Full-Circle ProtractorThink of the angle as a rotation of the minute hand on 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

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°.

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°.

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 where QP��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. The measure of angle PQR � 50°.

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Draw a 240° angle.

Step 1: Subtract: 360° � 240° � 120°.

Step 2: Draw a 120° angle.

The larger angle is a 240° reflex angle.

Measure each angle to the nearest degree.

1. 2. 3.

Draw each angle.

4. a 70° angle 5. a 280° angle 6. a 55° angleCheck your answers on page XXX.

Measure each angle to the nearest degree.

1. 2. 3.

Draw each angle.

4. a 70° angle 5. a 280° angle 6. a 55° angleCheck your answers on page 422.

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.

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Measurement

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 center of the baseline is over point A.

Step 4: Make a mark at 40° near the protractor. There are two scales on the protractor. Use the scale that makes sense for the size of the angle that you are drawing.

Step 5: Draw a ray from point A through the mark.

1. Into how many triangles can you divide each polygon?a. 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. How can you calculate

the number of triangles into which it can be divided without drawing a picture?

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 in the polygon by 180°.

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

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Measurement

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°.

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 six congruent angles.

The measure of one angle of a regular hexagon is 720°/6 � 120°.

regular hexagon (6 congruent sides and 6 congruent angles)

hexagon


Plotting Ordered Number Pairs

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

Every point on a rectangular coordinate grid can be named byan ordered number pair. The two numbers that make up anordered pair are called the coordinates of the point. The firstcoordinate 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 (2,4) names point A on the grid at the right. Thenumbers 2 and 4 are the coordinates of point A.

Measurement

Plot the ordered pair (4,2).

Locate 4 on the horizontal axis and draw a vertical line.

Locate 2 on the vertical axis and draw a horizontal line.

The point (4,2) is located at the intersection of the two lines.

The order of the numbers in an ordered pair is impor-tant. The ordered pair (4,2) does not name the samepoint as the ordered pair (2,4).

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

Locate (�4,3), (�4,�3), and (2,0).

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.

Draw a coordinate grid on graph paper. Plot the following points:1. (1,5) 2. (�4,�4) 3. (0,�3) 4. (3�

12�,�2)


234 two hundred thirty-four

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

10� degree. The

symbol for minutes is ( ' ).For example, the latitudeof Honolulu, Hawaii, is 21°18'N, or 21�

1680�

degrees north.

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Measurement

The latitude of the North Pole is 90°N. The latitudeof the South Pole is 90°S. The poles are the pointsfarthest 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.

Latitude and Longitude

Earth is almost a perfect sphere. All points on Earth are aboutthe same distance from its center. Earth rotates on an axis,which is an imaginary line that connects the North Pole andthe South Pole.

Reference lines are drawn on globes and maps to make placeseasier to locate. Lines that go east and west around Earth arecalled lines of latitude. The lines of latitude are often calledparallels because each one is a circle that is parallel to theequator. The equator is a special line of latitude. Every pointon the equator is the same distance from both the North Poleand the South Pole.

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 north to south. These are semicircles(half circles) that connect the two Poles. They are called linesof longitude or meridians.

The prime meridian is a special meridian that is labeled 0°.The prime meridian passes through Greenwich, near London,England. 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.

Meridians are not parallelsince they meet at the Poles.

The longitude of a place is measured in degrees. Lines west ofthe 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 latitude and longitude are both shown on a globeor a map, they form a pattern of crossing lines called a grid.The grid can help you locate any place on the map by simplynaming its latitude and longitude.

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Measurement

The longitude of Greenwich, England, is 0° becauseGreenwich lies on the prime meridian.

The longitude of Durban, South Africa, is 30°E.Durban is in the Eastern Hemisphere.

The longitude of Gambia (a small country in Africa)is about 15°W. We say that Gambia is 15 degreeswest of the prime meridian.

This map may be usedto find the approximatelatitude and longitudefor the cities shown.

For example, Denver,Colorado, is about 40°North and 105° West.

measurement - wikispaces to walk its length. ... length of a paper clip a man’s shoe length width...

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