SECTION 11.1 – LINEAR MEASUREMENT
History of U.S. Measurement
The English system of measurement grew
out of the creative way that people
measured for themselves. Familiar objects
and parts of the body were used as
measuring devices. For example, people
measured shorter distances on the ground
with their feet.
BOY…THAT’S ONE BIG FOOT!
In ancient times, the body ruled when it came to measuring. The length of a
foot, the width of a finger, and the distance of a step were all accepted
measurements.
Inch: At first an inch was the width of a man's thumb. In the 14th century,
King Edward II of England ruled that 1 inch equal 3 grains of barley placed
end to end lengthwise.
Hand: A hand was approximately 5 inches or 5 digits (fingers) across.
Today, a hand is 4 inches and is used to measure horses (from the ground to
the horse's withers, or shoulder).
Span: A span was the length of the hand stretched out, about 9 inches.
Foot: In ancient times, the foot was 111/42 inches. Today it is 12 inches,
the length of the average man's foot.
Yard: A yard was originally the length of a man's belt or girdle, as it was
called. In the 12th century, King Henry I of England fixed the yard as the
distance from his nose to the thumb of his out-stretched arm. Today it is 36
inches, about the distance from nose to out-stretched arm of a man.
Cubit: In ancient Egypt, a cubit was the distance from the elbow to the
fingertips. Today a cubit is 18 inches.
Lick: A Lick was used by the Greeks to measure the distance from the tip of
the thumb to the tip of the index finger.
Pace: The ancient Roman soldiers marched in paces, which were the length
of a double step, about 5 feet; 1,000 paces was a mile. Today, a pace is the
length of one step, 21/2 to 3 feet.
Unfortunately, these creative measuring devices allowed for different
measurements to be obtained when different people measured the same
items. Eventually, a standard was set so that all measurements represented
the same amount for everyone.
Standard Measures of Length
•
•
•
•
Inch = 0.083 feet
Foot = 12 inches
Yard = 3 feet or 36 inches
Mile = 5,280 feet or 1,760 yards
Conversions - Units of Length
Sometimes it is necessary to convert from one unit of measure to another
similar unit –from feet to inches, feet to yards, yards to feet, etc. The
following method is the easiest way to do this:
Look up the equivalent values for the two units and then decide whether to
multiply or divide.
The table below shows the standard equivalencies for length.
1 foot = 12 inches
1 yard = 3 feet or 36 inches
1 mile = 1760 yards or 5280 feet
HMMMM… DO I MULTIPLY? DIVIDE?
FIND OUT ON THE NEXT PAGE!
CONVERTING FROM A LARGER UNIT TO A SMALLER UNIT
When converting from a larger unit to a smaller unit, you need to multiply.
For example, to convert 8 feet to inches, look at the table of equivalencies to
find that 1 foot equals 12 inches. Multiply the number of feet (8) by the
number of inches in 1 foot (12) to find that 8 feet = 96 inches.
EXAMPLES:
0.2 miles = _____ feet
Answer: 1056 feet
Because we are converting from a larger unit to a smaller unit, we must
multiply the number of miles given (0.2) by the number of feet in one mile
(5280), so that 0.2 x 5280 = 1056 feet.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
18 yards = _____ inches
Answer: 648 inches
Because we are converting from a larger unit to a smaller unit, we must
multiply the number of yards given (18) by the number of inches in one yard
(36), so that 18 x 36 = 648 inches.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
NOW DO YOU GET IT? GOOD, ‘CAUSE THERE’S MORE!
CONVERTING FROM A SMALLER UNIT TO A LARGER UNIT
When converting from a smaller unit to a larger unit, you need to divide.
For example, to convert 48 feet into yards, look at the table of equivalencies
to find that 1yard equals 3 feet. Divide the number of feet (48) by the
number of feet in 1 yard (3) to find that 48 feet = 16 yards.
EXAMPLES:
219 feet = _____ yards
Answer: 73 yards
Because we are converting from a smaller unit to a larger unit, we must
divide the number of feet given (219) by the number of feet in one yard (3),
so that
219 ft
= 73 yards.
3 ft
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
64 inches = _____ yards
Answer: 1.78 yards
Because we are converting from a smaller unit to a larger unit, we must
divide the number of inches given (64) by the number of inches in one yard
(36), so that
64in
= 1.78 yards.
36in
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
SEE ATTACHED WORKSHEETS FOR MORE
PRACTICE PROBLEMS
The Metric System
Used in nearly every country in the world, the Metric System was devised by
French scientists in the late 18th century who recognized the need for a
standard unit of linear measurement. The goal of this effort was to produce
a system that did not rely on miscellaneous separate standards, and to use
the decimal system rather than fractions.
Today, the metric system is used for many different types of measurements
such as mass, area, volume, density, etc.; however, because the section
being covered deals with linear measurement, this lesson will focus on the
different units of length in the metric system.
Converting Metric Units
The metric system has prefix modifiers that are multiples of 10.
•
A kilometer (km) is 1000 meters
•
A hectometer (hm) is 100 meters
•
A decameter (dam) is 10 meters
•
A meter (m) is the basic unit of length
•
A decimeter (dm) is 1/10 meter
•
A centimeter (cm) is 1/100 meter
•
A millimeter (mm) is 1/1000 meter
As we move down the units, the next unit is one tenth as long. As we
move upward, each unit is 10 times as long.
It is possible to convert units by using the chart below. An easy way to
remember the set-up of the chart is by using the first letter of each unit in a
sentence such as “King Henry Danced Merrily Down Center Main”.
Conversions among metric lengths are accompanied by multiplying or dividing
by powers of ten as shown on the “steps”, however, there is a much simpler
way of figuring conversions, but you must learn the chart below.
10 3
kilo
10 2
hecto
10 1
deka
10
meter
10 −1
deci
10 −2
10 −3
centi
milli
Step 1:
What is being converted? Go to that “stair-step” on the chart.
Step 2:
What are you converting to? Move up or down the stair-steps.
Step 3:
Count the number of stair-steps you moved.
Step 4:
If you moved DOWN the stair-steps, move the decimal point to
the RIGHT that many steps.
If you moved UP the stair-steps, move the decimal point to the
LEFT that many steps.
EXAMPLES:
Using the chart above, convert the following:
1.4 km = _____ m
Answer: 1400 m
Because we moved 3 steps down from km to m, we moved the decimal point
3 spaces to the right.
285 mm = _____ m
Answer: 0.285 m
Because we moved 3 steps up from mm to m, we moved the decimal point 3
spaces to the left.
0.03 km = _____cm
Answer: 3000 cm
Because we moved 5 steps down from km to cm, we moved the decimal
point 5 spaces to the right.
SEE ATTACHED WORKSHEETS FOR MORE
PRACTICE PROBLEMS
Distance Around a Plane Figure
The perimeter of a simple closed curve is the distance around the figure. In
a regular polygon, it is the sum of the lengths of its sides.
To find the perimeter of a
regular polygon like in the
examples shown below, add
together the lengths of all
of the sides of the figure.
Examples:
4 cm
4 cm
4 cm
Perimeter: 4cm + 4cm + 4cm = 12cm
8 cm
3 cm
3 cm
8 cm
Perimeter: 3cm + 8cm + 8cm + 3cm = 22cm
6 cm
7 cm
7 cm
3 cm
Perimeter: 7cm + 6cm + 3cm + 7cm = 23cm
2cm
2cm
2cm
2cm
2cm
2cm
Perimeter: 2cm + 2cm + 2cm + 2cm + 2cm + 2cm = 12cm
See attached
worksheets for
practice perimeter
problems
Circumference of a Circle
The circumference of a circle is the distance around a circle, also known as
the perimeter of a circle.
Before determining the circumference of a circle, either the radius or the
diameter of the circle must be known.
The radius of a circle is the distance of the center of a circle to any point on
the circumference.
The diameter of a circle is a straight line passing through the center of a
circle that touches both sides of the circumference.
About π (pi):
The ancient Greeks discovered that if they divided the circumference of any
circle by the length of its diameter, they always came up with approximately
the same number (3.14).
Today, the ratio of circumference C to diameter d is symbolized as
π (pi).
The approximate values of π (pi) are
22
1
, 3 , or 3.14
7
7
Formulas for determining the circumference of a circle:
If radius is given, use C = 2πr
(circumference = 2 times pi times radius)
If diameter is given, use C = πd
(circumference = pi times distance)
EXAMPLES:
Problem:
Find the circumference of a circle if the radius is 2m.
Solution:
C = 2πr
= (2)(3.14)(2m)
= 12.56 m
Problem: Find the circumference of a circle if the diameter is 6cm
Solution: C = πd
= (3.14)(6cm)
= 18.84 cm
Problem: Find the circumference of a circle if the radius is 3.5cm
Solution: C = 2πr
= (2)(3.14)(3.5cm)
= 21.98 cm
SEE ATTACHED WORKSHEETS FOR MORE
PRACTICE PROBLEMS
THE END…FINALLY!
Circumference of Circles
Find the Circumference for each.
Assume = 3.14
1.
2.
g = 20 yd
m = 46 cm
________________
________________
3.
4.
s = 11 ft
e = 4 cm
________________
________________
5.
6.
s = 10 cm
m = 83 m
________________
________________
8.
7.
g = 13.9 in
e = 26.8 mi
________________
________________
Perimeter
Find the perimeter.
2.
1.
v = 3 ft
t = 7 ft
r = 12 ft
s=t
e=7m
f = 13 m
_________________
________________
3.
4.
a = 3 ft
c = 5 ft
b=c
The side d of this square is
39 yd
________________
________________
6.
5.
m = 14 yd
All sides are equal
All sides equal 7 yd
________________
7.
v = 4 ft
t = 6 ft
r = 12 ft
s=t
________________
8.
The side d of this square is
21 m
________________
________________
10.
9.
All sides equal 11 cm
________________
e=9m
f = 17 m
________________
Metric Conversions
Convert the following.
1.
400009 cm = ___________ km
2.
0.00069 km = ____________ mm
3.
828 m = _____________mm
4.
0.125 m = _____________cm
5.
276 km = _____________m
6.
0.26678 m = ____________km
7.
9.651 cm = ____________mm
8.
84322 dm = ___________ m
9.
6.00042 dam = ___________cm
10.
534.255 cm = ____________ mm
11.
00254.32 mm = ____________ hm
12.
82,800 cm = ____________ m
13.
426,000,000 mm = ____________ km
14.
0.864 m = ____________ mm
15.
8 cm = ____________m
Standard Conversions
Convert the following.
1.
3.5 miles = ____________ feet
2.
9 yards = ____________ inches
3.
84 inches = ____________ feet
4.
120 feet = ____________ inches
5.
4.2 miles = ____________ yards
6.
24 yards = ____________ feet
7.
6.2 feet = ____________ inches
8.
1.2 miles = ____________ feet
9.
15840 feet = ____________ miles
10.
255 inches = ___________ feet
11.
42 feet = ____________ yards
12.
6 miles = ____________ yards
13.
32 yards = ____________ inches
14.
44 yards = ____________ miles
15.
72 feet = ____________ inches
Answers to Worksheets
Circumference
1.
2.
3.
4.
5.
6.
7.
8.
62.8 yd
288.88 cm
69.08 ft
25.12 cm
62.8 cm
521.24 mi
43.646 in
168.304 mi
Perimeter
1. 40 m
2. 29 ft
3. 156 yd
4. 13 ft
5. 21 yd
6. 84 yd
7. 34 ft
8. 84 m
9. 33 cm
10. 52m
Metric Conversions
1. 4.00009 km
2. 690 mm
3. 828000 mm
4. 12.5 cm
5. 276000 m
6. 0.00026678 km
7. 96.51 mm
8. 843220 m
9. 6000.42 cm
10. 53425.5 mm
11. .0025432 hm
12. 828 m
13. 426 km
14. 0864 mm
15. .008 m
Standard Conversions
1. 18480 ft
2. 324 in
3. 7 ft
4. 1440 in
5. 73.92 yd
6. 72 ft
7. 74.4 in
8. 6336 ft
9. 3 mi
10. 21.25 ft
11. 14 yd
12. 10560 yd
13. 1152 in
14. 0.025 mi
15. 864 in
NOTES AND ACTIVITES FOR THIS LESSON WERE
PREPARED AND PRESENTED BY:
CHRISTI L. ALARCON
JODIE CORDOVA
LYZA DE HOYOS
MARIA ELENA GUTIERREZ
REBECCA IDROGO
Fundamentals of Math II
March 25, 2004