Section 10.1 Systems of Measurement
ENGLISH UNITS
Length As societies evolved, measures became more complex. Since most units of measure had developed
independently of one another, it was difficult to change from one unit to another. The English system, for
example, arose from a hodgepodge of nonstandard units: The inch was the length of 3 barleycorns placed end
to end, the foot was the length of a human foot, and the yard was the distance from the nose to the end of an
outstretched arm. In the twelfth century, the yard was established by royal decree of King Henry I of England
as the distance from his nose to his thumb. Gradually, the English system of measurements was standardized.
The common units for length are shown below.
Volume There are two
methods of measuring volume
in the English system. One
uses cubes whose edges have
lengths of 1 inch, 1 foot, or 1
yard. For example, a cubic inch
is the volume of a cube whose
edges are each 1 inch long.
One gallon is equal to 231 cubic inches. Determine the number of cubic inches in each of the following volume measures.
1. 1 quart
2. 1 cup
Weight The English system has two systems for measuring weight: one for precious metals, in which there are 12
ounces in a pound (troy unit); and one for everyday use, in which there are 16 ounces in a pound (avoirdupois unit).We
will use the avoirdupois unit.
English Units for Weight
1. 14.3 pounds equals how many ounces?
ounce oz
pound
pound lb 16 ounces
ton
tn 2000
pounds
2. 3200 pounds equals how many tons?
Converting units in the English System
Use the rhyme, “big to small, multiply all. Small to big, divide the pig.”
Convert each of the following units in the English System. Us the conversion sheet if you need to.
1) 15,840 ft = ___________ mi
2) 18.75 ft = ____________ in
3)
1 mi = ____________ in
4) 36 oz = ___________ lbs
5) .012 T = ____________ oz
6)
2 c = ____________ qt
7. A curtain for a single window can be made from a piece of material that is 1 yard wide and 48 inches long. Suppose
you need two curtains per window and have six windows. If the curtain material comes in rolls 1 yard wide, how many
yards of length will be needed to make curtains for all six windows?
METRIC UNITS
In 1790, in the midst of the French Revolution, the metric system was developed by the French Academy of Sciences. To
create a system of “natural standards,” the scientists subdivided the length of a meridian from the equator to the north
pole into 10 million parts to obtain the basic unit of length, the meter.
Once a basic unit is established in the metric system, smaller units are obtained by dividing the basic unit into 10, 100,
and 1000 parts. Larger units are 10, 100, and 1000 times the basic unit. These units are named by attaching prefixes to
the name of the basic unit.
The Prefixes are
Prefix
Symbol
Factor Number
Factor Word
Kilo
k
1,000
Thousand
Hecto
h
100
Hundred
Deka
da
10
Ten
Deci
d
0.1
Tenth
Centi
c
0.01
Hundredth
Milli
m
0.001
Thousandth
One way to convert units in the metric system is to use the acronym “king henry died by drinking chocolate milk”.
Kilo hecto deka base deci centi milli
Use the “KING HENRY” method to convert units in metric system
1) 245 mg = ___________ cg
2) 2.3 dl = ____________ kl
3)
.023 km = ____________ cm
4) 134 cm = ___________ hm
5) .19 dag = ____________ cg
6)
12.3 cl = ____________ ml
7. The recipe for a fruit punch calls for these ingredients: 3.5 liters of unsweetened pineapple juice, 400 milliliters of
orange juice, 300 milliliters of lemon juice, 4 liters of ginger ale, 2.5 liters of soda water, 500 milliliters of mashed
strawberries, and a base of sugar, mint leaves, and water that has a total volume of 800 milliliters.
a. How much punch will the recipe make in liters?
b. If you serve the punch at a party of 30 people, how many milliliters of punch will there be per person?
c. This punch was sold at a fair, and each drink of 80 milliliters cost 25 cents. What was the profit on the sale of
this punch if the ingredients cost $12.50 and all the punch was sold?
8. A car owner has her tank filled and notices that the odometer reads 14,368.7 kilometers. After a trip in the country, it
takes 34.5 liters to fill the tank, and the odometer reads 14,651.6. How many kilometers per liter is this car getting?
Cubic units
Notice how convenient it is to change from one unit of volume to another. Since 1 liter equals 1000 milliliters, the number
of milliliters in 1.3 liters is 1.3 × 1000. Similarly, to change from 245 milliliters to liters, we divide by 1000. With metric units,
conversions can be done mentally by multiplying and dividing by powers of 10.
A liter is the volume of a cube whose sides each have a length of 10 centimeters as in the diagram below. Such a cube is
called a cubic decimeter. The dimensions of the small cube in part b are each 1 centimeter, and this cube is called
a cubic centimeter.
Solve the following volume problems.
1. 45 cubic centimeters equals how many milliliters?
2. 1.35 liters equals how many cubic centimeters?
3. 800 cubic centimeters equals how many liters?
Converting units of speed and time
1. How fast is a car going in miles per hour when it is traveling 88 feet per second?
2. A snail can crawl at a speed of .013 m/s. How fast is it going in mi/hr?
3. It takes a monkey 2 hours to eat 14 frogs, how many cats could the monkey eat in 1 hour?
Section 10.2 Area & Perimeter Part 1
Perimeter refers to the sum of the lengths of the sides of a figure.
Area generally refers to the number of square units needed to completely fill the interior of a
two-dimensional figure.
Example: Find the area of the figure illustrated on the square lattice. Use the unit square
shown as the fundamental unit of area.
STANDARD UNITS OF AREA
To standardize area measurement, a square became the accepted area unit shape. However, the size of the unit square
differed in the two predominant systems of measurement: the English system and the metric system.
English Units In the English system, area is measured by using squares whose sides have lengths of 1 inch, 1 foot, 1
yard, or 1 mile. Each square unit is named according to the length of its sides. The 1 inch × 1 inch square below has an
2
area of 1 square inch, abbreviated 1 sq in or 1 in (think of the exponent 2 as indicating a square).
The different square units are related to one another.
1. How many square inches equal 1 square foot?
2. How many square feet equal 1 square yard?
Metric Units In the metric system there is a square unit for area corresponding to each unit for length. For example,
2
a square meter (1 m ) is a square whose sides have a length of 1 meter. Square meters are used for measuring the
areas of rugs, floors, swimming pool covers, and other such intermediate-size regions. Smaller areas are measured in
2
square centimeters. A square centimeter (1 cm ) is a square whose sides have lengths of 1 centimeter.
Determine the following relationships between the metric units for area.
1. How many square millimeters are there in 1 square centimeter?
2. How many square centimeters are there in 1 square meter?
PERIMETER
Another measure associated with a region is its perimeter—the length of its boundary. The perimeter of the figure
below is 23 centimeters, which is greater than the width of this page.
Each of the following figures has an area of 4 square centimeters. What is the perimeter of each figure?
For each figure determine the area in square centimeters and the perimeter in centimeters.
AREAS OF POLYGONS
Rectangles Rectangles have right angles and pairs of opposite parallel sides, so unit squares fit onto them quite easily.
The rectangle in the figure below can be covered by 24 whole squares and 6 half-squares. Its area is 27 square units.
This area can be obtained from the product 6 × 4.5, because there are
squares in each of 6 columns.
Find the area and perimeter of each figure.
Parallelograms Fitting unit squares onto a figure is a good way for schoolchildren to acquire an understanding of the
concept of area. One of the basic principles in finding area is that a region can be cut into parts and reassembled without
changing its area. This principle is useful in developing a formula for the area of a parallelogram.
Formula =
Find the area of each parallelogram below.
Triangles The triangle below is covered with 1 centimeter × 1 centimeter squares and parts of squares. Can you see
why this shows that the area of the triangle is more than 4 square centimeters? Since it is inconvenient to cover the
triangle with squares, we will use a different approach to find its area. Two copies of a triangle can be placed together to
form a parallelogram, as shown in part b. This can be accomplished by rotating the triangle in part a about the midpoint of
side
. This can be done with every triangle.
As long as the base and height are perpendicular, you can use them to find the area.
Formula =
Find the area of each triangle below.
Trapezoids It is inconvenient to cover a trapezoid with square units because of its sloping sides. However, as with a
triangle, we can obtain a parallelogram by placing two trapezoids together.
Find the area of the trapezoid below.
Formula =
Section 10.2 Area & Perimeter Part 2
CIRCUMFERENCES AND AREAS OF CIRCLES
Circles are part of our natural environment. The Sun, the Moon, flowers, whirlpools, and cross sections of some trees
have circular shapes. School children see many everyday examples of circles, such as bottle caps, tops of cans, lamp
shades, and toys like the hula hoop.
In small groups measure the circumference and diameter of a circle. Have another student
check your answer. Take careful measurements – precision and accuracy are important
here! The unit of measure you use is irrelevant as long as you are consistent and precise.
Record your results and the results of several other classmates on the chart below.
Circumference
Circumference

Diameter
Circumference of a Circle (Perimeter) =
Diameter
Ratio (C/d)
There are several ways of obtaining the formula for the area of a circle, one way is shown below.
Make sure your calculator is on DEG and use 3.14 for pi.
Formula for the area of a circle =
Find the area of each circle.
Find the area of each shaded region.
Section 10.3 Volume Part 1
English Units Cubic units are named according to the length of their edges. The most commonly used units for
measuring non liquid volume in the English system are the cubic inch, the cubic foot, and the cubic yard. The cubes for
these units are illustrated below. These cubes are not drawn to scale.
Determine the following English unit relationships.
1. How many cubic inches equal 1 cubic foot?
2. How many cubic feet equal 1 cubic yard?
3. How many cubic inches equal 1.4 cubic feet?
Metric Units The common metric units for measuring non liquid volume are the cubic millimeter, the cubic
centimeter, and the cubic meter. Each unit is named according to the length of the edges of its cube. For example, the
edges of th e cube for the cubic centimeter each have a length of 1 centimeter.
VOLUMES OF SPACE FIGURES
To find the volume of a space figure, think of taking its
base area times its height. That would fill it up. Look at
the figure to the right.
The same is true for any other prism, even triangular
prisms. Find the area of the base, then multiply it by its
height.
Formula =
A cylinder is no different. Find
the area of the base and
multiply it by its height.
Formula =
The formula for the volume of an oblique prism is suggested by beginning with a stack of cards, as in the Figure below
and then pushing them sideways to form an oblique prism, as in part B. Notice that the volume stays the same, its still 52
cards. Therefore the formula is the same as it is for a parallelogram base.
Volume of pyramid =
Volume of a cone =
Volume of a sphere =
Find the volume to the nearest tenth of a unit.
Section 10.3 Surface area Part 2
SURFACE AREAS OF SPACE FIGURES
The surface area of a prism is the sum of all of its faces. To find the surface area of a prism, you can find each separate
polygon that make up its individual faces or you can use a formula. I will show you both ways now.
Cylinders are a little different because they have the curved lateral surface. Take a look at the figure
below.
You can see how this relates to the formula for the surface area of a cylinder =
Find the surface area of the
cylinder.
Surface areas of pyramids =
and cones =
Surface areas of spheres =
Find the surface area of the sphere below. Convert to cubic inches when done.