UNIT ONE: Prealgebra in a Technical World
1.6 Measurements: Lengths & Areas
1. Find perimeters and missing lengths.
SWBAT
2. Use area formulas for triangles and special quadrilaterals.
Geometry, the relationship of shapes in space, is still every human being’s first use of
math. For instance, whenever we walk, we notice shapes in space. A hunter needs to know
shapes in space! As early humans formed social clusters, they counted to divide the spoils of
the hunt, and arithmetic was born. As soon as these clusters of people began to build, they
combined arithmetic and geometry in order to measure.
In this chapter we study and solve problems using measures of lengths and areas.
Lengths and Perimeters of Shapes
The first measuring units used were the units that were readily at hand. These units
were lengths on the human body. For instance, a “cubit” is the length from the tip of your
elbow to the tip of your fingers. When individual craftsmen finished each project on their own,
personal measurements were perfectly acceptable.
As civilization grew, measurements needed to be standardized. King Edward I (r 12721307) of England first standardized measures which became our yard, foot and inch in the 13 th
century.
In the U.S., we still use a modified British system of measure. This makes the U.S. one of
only three countries that have not adopted the metric system as their official measuring
system. Myanmar and Liberia are the other two non-metric
nations. King Edward’s yard, foot, and inch are measures of
length. We can add, subtract, multiply, and divide measures of
length, and we often measure the length of sides of a shape.
Table 1
U.S. Length Conversions
1 foot = 12 inches
1 yard = 3 feet = 36 inches
1 mile = 5,280 feet
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SECTION 1.6 Measurements: Lengths & Areas
The perimeter is the distance around a shape. When working with measurements, we must
use the same units.
Table 1 gives length conversions that we use most often in the U.S. system.
Example 1: What is the perimeter of this triangle?
1 yard
2 feet
4 feet
Think it through: We have two different units. We change 1 yard to 3 feet, then we can add
2 ft + 4 ft + 3 ft.
ANSWER: The perimeter is 9 feet and, since there are 3 feet in a yard, the perimeter is
also 3 yards.

Check Point 1
What is the perimeter of the rectangle at right:
________________________________________________ 1 ft
________________________________________________
54 inches
Area Formulas for Shapes
Area is the measure of the surface of a flat shape. We find the area by determining the
number of square units that fill the shape.
All area formulas are related to the area of
a rectangle, and all areas are measured in square
units.
The rectangle on the left has been filled
with squares that are 1 cm on each side. Notice
that we have 5 rows of 6 squares in each row. To
find the total number of squares, we multiply
length and width: 5𝑐𝑚 ∙ 6𝑐𝑚 = 30𝑐𝑚2 .
UNIT ONE: Prealgebra in a Technical World
FORMULA: For a rectangle, the area is given by 𝑨 = 𝒍 ∙ 𝒘 where 𝒍 is the
length and 𝒘 is the width of the rectangle.
Table 2
Area is recorded in square units. Table 2 gives you
U.S. Area Conversions
the area conversion facts that we use most often in
the U.S. system. These facts are also available online
and in a dictionary. Perhaps they are even on your
1 ft2
=
1 yd2 =
12 in x 12 in
=
144 in2
3 ft x 3 ft
=
9 ft2
cell phone!
Before we discuss areas of special four-sided
shapes, we should define our shapes. You know two
of the area formulas already.
1 acre = 43,560 ft2
1 square mile = 2,878,400 ft2
AREA OF A TRIANGLE:
To develop the formula for the
area of a triangle, we start with a triangle.
We have drawn four different triangles on
the right. You may draw your own and
follow along with the instructions.
1. To make it easier to find the
height, select the longest side of
your triangle to be the base.
2. Next draw a height that is at a right angle from this base and ends at the opposite vertex.
3. Using the base and the height, draw a
rectangle around your triangle. We have
done this on all four triangles below:
The original triangles take up exactly half of
the area of the rectangles we drew around them.
The base and height of each triangle are the
length and width of the rectangles we drew.
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SECTION 1.6 Measurements: Lengths & Areas
Thus the area of a triangle is ½ (half) the area of the rectangle constructed with the base
and height.
1
FORMULA: The area of a triangle is given by 𝐴 = 2 (𝑏 ∙ ℎ) where 𝑏 is the
base and ℎ is the height. Each of the following are equivalent to the
1
𝑏ℎ
formula above: 𝐴 = 𝑏ℎ ÷ 2 , 𝐴 = 2 𝑏ℎ and 𝐴 = 2 .
Example 2: Find the area of each of the shapes. (Hint: Draw a diagram like step 3 above.)
a. A rectangle with a length of 7 inches and a width of 4 inches.
b. A triangle with a base of 4 inches and a height of 7 inches.
c. The triangle at right.
ANSWER: a. 𝐴 = 𝑙 ∙ 𝑤 = 7 ∙ 4 = 𝟐𝟖 𝒊𝒏𝟐
1
b. 𝐴 = 2 (𝑏 ∙ ℎ) =
1
c. 𝐴 = 2 (𝑏 ∙ ℎ) =

1
2
1
2
(4 ∗ 7) = 𝟏𝟒 𝒊𝒏𝟐
(4 ∙ 7) = 𝟏𝟒 𝒊𝒏𝟐
Check Point 2
Find the area of each of the shapes. (Hint: Draw a diagram.)
a. A rectangular room that is 15 ft by 12 ft.
a. ______________
b. A triangle with a base of 15 cm and a height of 12 cm.
b. ______________
c. A triangle with a base of 12 in and a height of 15 in.
c. ______________
UNIT ONE: Prealgebra in a Technical World
AREA OF A PARALLELOGRAM:
Figure 1 shows a parallelogram, a four-sided
Figure 1
shape with opposite sides parallel. The formula for
the area of a parallelogram is also related to the
formula for a rectangle.
Figure 2
We label a parallelogram with its base and height in
Figure 2. The height of any shape is always at a right
angle to its base.
Figure 3
In Figure 3 we determine how the area of a
parallelogram is related to the area of a rectangle
by cutting off the shaded triangle and attaching
it to the opposite side.
By creating this rectangle from a parallelogram, we see that the area of any
parallelogram is the the product of the parallelogram’s base and height.
FORMULA: For a parallelogram, the area is given by 𝐴 = 𝑏ℎ,
where b is the base and h is the height.
TRAPEZOIDS:
Trapezoids are quadrilaterals with two parallel sides. We call
these sides “bases.” In Figure 4 the bases are labeled b1 and b2. We
use b1 and b2 because a trapezoid has two bases. It does not matter
which base we call b1 and which we call b2; the area will be the
same. The height of any shape can be labeled “h.”
Every trapezoid can be divided into two trapezoids that are
Figure 4
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SECTION 1.6 Measurements: Lengths & Areas
each ½ of the original height.
If we rotate the smaller trapezoid and bring it alongside the larger one, we create a
parallelogram.
The base of this parallelogram is the
sum of the original bases, b1 + b2, and the
height is half of the original height of the
trapezoid.
1
FORMULA: For a trapezoid, the area is given by 𝐴 = 2 (𝑏1 + 𝑏2 )ℎ, where 𝑏1
and 𝑏2 are the bases and ℎ is the height. The formulas 𝐴 = ℎ(𝑏1 + 𝑏2 ) ÷ 2
and 𝐴 =
ℎ(𝑏1 +𝑏2 )
2
are equivalent formulas for the area of a trapezoid.
Example 3: Identify the base and height for each figure. Then find the area of the given
shapes.
8 yards
5 yards
a.
18 yards
Think it through: The shape is a parallelogram. The base is 18 yards and the height is 5
yards. The length of 8 yards does not matter when we find the area. The
formula for the area is 𝑨 = 𝒃𝒉 so 𝑨 = (𝟏𝟖) ∙ (𝟓) square yards.
ANSWER: 𝟗𝟎 yds2
b.
6 cm
15 cm
18 cm
20 cm
UNIT ONE: Prealgebra in a Technical World
Think it through: The shape is a trapezoid. The bases are 18 cm and 6 cm and the height is
15 cm. The length of 20 cm does not matter when we find the area. The
𝟏
𝟏
formula for the area is 𝑨 = 𝟐 (𝒃𝟏 + 𝒃𝟐 )𝒉 so 𝟐 (𝟏𝟖 + 𝟔)𝟏𝟓 square cm.
ANSWER: 𝟏𝟖𝟎 cm2

Check Point 3
Identify the base (or bases) and height for each figure. Then find the area of the given shapes.
a.
b.
2 in
6 yards
10 in
16 in
4 yards
_______________________
_________________________
Example 4: A deck has the given dimensions. What is the area?
18 feet
~ 23 feet
9 feet
5 feet
40 feet
Think it through: We can divide this deck into a rectangle and a trapezoid two different ways.
Or we can find the area of the large rectangle that surrounds the whole
deck and subtract the area of the triangle that is “cut off” the edge of the
deck. We choose this third idea.
9
f
e
e
t
1
8
1
~
~
8
9
2
2
4 2 5
4 2
f 0
f f 0 f
f
rectangles
and trapezoids
e
e e
e f
e
e f
e f
e
e e
e
t e
t e
t
t e
t
e
e
t
t
t
9
5
f
e
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t
f
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t
1
8
4
f 0
e
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t e
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t
~
2
2
f
e
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t
Rectangle
& triangle
5
f
e
e
t
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SECTION 1.6 Measurements: Lengths & Areas
The area of the rectangle is 9 ft ∙ 40 ft = 360 ft 2 .
1
1
The area of the triangle is 2 (40 ft − 18 ft)(9 ft − 5 ft) = 2 (22 ft)(4 ft) = 44 ft 2 .
The difference in area is 360 ft 2 − 44 ft 2 = 316 ft 2 .
ANSWER: The area of the deck is 𝟑𝟏𝟔 𝐟𝐭 𝟐 .

Check Point 4
Marilyn is planting her prize-winning dahlias
14 feet
in a garden bed between walkways. The
10 feet
shape of the bed is shown on the right. Her
4 ft
fertilizer instructions say she should use one
20 feet
tablespoon of fertilizer for every 10 square
feet.
a. What is the area of the bed in square feet? ____________________________
b. About how many tablespoons of fertilizer should she use? ________________
Your goal is to know, and know how to use, the formulas you have studied in this section.
Create your own “quick list” by labeling each shape and write the formula for its area in the
table below. Better yet make flash cards to memorize the shapes and area formulas.
Triangles
Squares
Rectangles
Parallelograms
Trapezoids
𝑙
𝑤
A=
A=
𝐴 = 𝑙∙𝑤
A=
A=
UNIT ONE: Prealgebra in a Technical World
1.6 Exercise Set
Name _______________________________
Skills
Find the perimeter of the following:
1. a square with a side of 29 inches
2. a square with a side of 11 yards
3. a rectangle with a length of 42 feet and a
width of 15 feet
4. a rectangle with a width of 31 meters and
a length that is twice the width
5. a parallelogram with sides that measure
14 feet and 11 feet
6. a parallelogram with sides of 19
centimeters and 41 centimeters
7. a triangle with sides of 3 feet, 5 feet, and
48 inches
8.
9. a trapezoid with bases 9 meters and 18
meters and sides 4 meters and 5 meters
10. a trapezoid with bases 12 meters and 15
meters and two equal sides of 10 meters
a triangle with sides of 3 yards, 9 feet and
6 feet
Find the area of the following:
11. a square with a side of 39 inches
12. square with a side of 25 yards
13. a triangle with a base of 48 inches and
height of 1 foot
14. a triangle with a height that is one-half of
its base of 16 inches
15. a parallelogram with height of 3 feet and
base of 36 inches
16. a parallelogram with base of 420 yards
and height of 120 yards
17. a rectangle that is 25 feet by 32 feet
18. a rectangle that is 14 meters by 29
meters
19. a trapezoid with bases 15 yards and 11
yards and height of 6 yards
20. a trapezoid with height of 26 feet and
bases 10 feet and 15 feet
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80
SECTION 1.6 Measurements: Lengths & Areas
For problems 21 to 24 use this information: A five equal-sided figure is a pentagon. A six equalsided figure is a hexagon. An eight equal-sided figure is an octagon.
21. A pentagon measures 27 meters on a
side. What is the perimeter of the
pentagon?
22. An octagonal building measures 24
meters on a side. What is the perimeter
of the building?
23. A flower bed shaped like a hexagon
measures 8 feet on a side. How much
fencing will you need to buy to surround
the flower bed?
24. The Pentagon in Arlington, Virginia,
measures approximately 921 feet on a
side. What is the distance around the
Pentagon?
Applications
25. A homeowner wants to fence her
property. She measures the width and it
is 104 feet. The length is 256 feet. How
much fencing does she need to buy?
26. A builder wants to sod the yard at a new
home he has built. The yard measures
45 feet by 65 feet. Sod costs $0.14 per
square foot. How much will the sod
cost?
27. Luiz is putting laminate flooring in his master
bathroom. The bathroom is 12 feet by 9 feet,
with a corner cut out for a 4 feet by 3 feet
shower. How many square feet of laminate
does he need to buy?
28. Lionel is setting up a model railroad using a 4
feet by 8 feet sheet of plywood. He is cutting
a 2 foot by 4 foot rectangle out of the middle,
so he can work on the layout and watch the
train go around him. How many square feet
of surface will he have left to set up his
railroad?
29. Lionel (from the previous problem), would like
to cut out a circle with a 3 foot diameter,
instead of a rectangle, but he doesn't
remember how to calculate the area of a
circle. He decides to estimate by using the
area of a square formula. Approximately how
many square feet will he have to place
decorative items on his train board, if he cuts
out a circle?
UNIT ONE: Prealgebra in a Technical World
30.
A laundry room has to be modified to hide
exposed plumbing that was recently installed. The
original floor plan of 12 feet by 8 feet was altered
by a short wall that was placed in one corner of the
room. The short wall was constructed using a
triangular base of 2 feet and height of 1 foot.
a. What was the original square footage of the room?
b. What is the square footage after the room was modified?
31.
Wolfgang has a rectangular pool that is 20 feet by 30 feet. He wants to pour a 4 foot wide
sidewalk around the entire pool. Draw a picture and find the following:
a. What is the perimeter of the pool?
b. What is the outside perimeter of the sidewalk?
c. What is the area of the rectangular pool?
d. What is the area of the sidewalk?
32. Amanda has a rectangular flower bed that is 15 ft by 20 ft. She is going to install a 3 foot wide
sidewalk around this flower bed. Draw a diagram to find the following:
a. What is the perimeter of the flower bed?
b. What is the outside perimeter of the sidewalk?
c. What is the area of the rectangular pool?
d. What is the area of the sidewalk?
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SECTION 1.6 Measurements: Lengths & Areas
Review and Extend
33.
Rudy is purchasing bamboo flooring for both his dining room and great room. He is
calculating the square footage using the house plans. Use the following drawing to
answer the questions. (Hint: Notice that some lines are drawn to split these shapes.)
a. What is the square footage of the dining
room whose floor plan is drawn on the
right?
b. What is the exact square footage of the
great room whose diagram is drawn on
the right?
c. If the bamboo flooring that Rudy wants
costs $4.89 a square foot, what is the
approximate amount that Rudy will
spend for his floor covering?
(Hint: Round up! Answers will vary, but
yours should be close.)
34. IN WRITING: What does UPS stand for? Why do we think about more than S (solve)
when solving math problems? (Use your own paper for your response.)
35. Estimate by rounding. Show how you rounded the expression and give your answer.
a. 389 + 1,213 b. 1,213 – 380
c. 38 ∙ 123
d. 123 ÷ 38
36. Which problem solving strategy is used to complete: 1, 2, 6, 12, 20, ____, ____, _____ ?
37. Calculate exactly:
a. 389 + 1,213
b. 1,213 − 389
c. 39 ∙ 125