Section 3.4
Quadrilaterals
Pre-Activity
Preparation
Interesting geometric shapes and patterns are all around us when we start looking for them. Examine a row of
fencing or the tiling design at the swimming pool. Notice how squares, rectangles, parallelograms and other
plane geometric figures combine to offer texture, interest, and important geometric structure in our lives.
Modern and ancient designs in art make use
of simple geometric shapes put together in
complicated patterns. Tiling and mosaics
use little squares of material to fit a pattern or
make a picture. The architectural mosaic on
the outside of the Muhammad Ali Center in
Louisville, Kentucky (at right) is constructed
of ceramic tile rectangles of the same size but
different colors to form images of Muhammad
Ali, the world-famous boxer. Modern
encrypting software uses tessellations and digital imaging to protect our privacy by sliding and rotating
regular shaped polygons in a predictable pattern that can be coded or decoded. For more information look up
the bold-faced words in any Internet search engine.
Learning Objectives
• Find the perimeter of a quadrilateral
• Find the area of a quadrilateral
Terminology
Previously Used
side
New Terms
to
Learn
quadrilateral
triangle
207
Chapter 3 — Geometry
208
Building Mathematical Language
Quadrilaterals
In the last two sections, we used formulas for finding
the perimeters and areas of triangles and circles.
This section concerns the next set of basic shapes:
quadrilaterals—closed plane figures with four
sides.
closed
not closed
Following are profiles of five basic quadrilateral shapes, including their defining characteristics, perimeter,
and area formulas, and observations or clarifying comments.
s
Square
s
A four-sided figure with
all four sides equal. Each
internal angle measures
90°
s
s
Perimeter (P)
Area (A)
P = 4s
A = s2
Perimeter equals four
times the length of
one side
Area equals side
squared
OBSERVATIONS: Area measurements use “square
units”—one square unit is a square that is one unit
(1 inch, foot, meter, mile, etc.) long on each side:
Rectangle
l
A four-sided figure with
opposite sides equal and
parallel. Each interior
angle measures 90°
w
w
l
Perimeter (P)
P = 2l + 2w
or
Area (A)
A = lw
P = 2(l+w)
Perimeter equals
twice the length plus
twice the width
Area equals length
times width
OBSERVATIONS: We have used length multiplied
by width to describe multiplication—the dimensional
measurements give context to finding the product.
Parallelogram
b
B
w
A
C
w
h
b
D
A four-sided figure with
opposite sides equal and
parallel. The interior angles
are NOT necessarily equal
to 90°
Perimeter (P)
Area (A)
P = 2b + 2w
A = bh
Perimeter equals twice
the length (base) plus
twice the width
Area equals base
times height
OBSERVATIONS: There are three important measurements for a parallelogram: base (length), side
(width), and height (altitude). Height is required to find
the area.
Trapezoid
s3 (b2)
s2
h
s4
A four-sided figure with
two unequal parallel sides
and two non-parallel sides
s1 (b1)
Perimeter (P)
Area (A)
P = b1+b2+s3+s4
A=½h(b1+b2)
Perimeter equals the
sum of the lengths of
each side
Area equals onehalf times the height
times the sum of the
bases
OBSERVATIONS: Many rooftops are trapezoidal in
shape.
Section 3.4 — Quadrilaterals
s
B
s
A
Rhombus
C
s
h
s
209
D
A four-sided figure with all
sides equal. Its opposite
sides are parallel.
Perimeter (P)
Area (A)
P = 4s
A = sh
Perimeter equals four
times the length of
one side
alternatively:
A = ½ d1d2
B
d2
A
C
d1
D
OBSERVATIONS: A rhombus is sometimes called a
diamond shape; think of a kite or a baseball infield.
Can you use the area formula
for triangles to prove that:
1
Arearhombus = d1d 2
2
given that the diagonals of a rhombus bisect (cut in half)
each other at right angles?
Baseball Diamond
or Baseball Square?
Orientation
is
important
in determining the name
of a figure. Even though
an infield is square, the
orientation makes it look
diamond shaped. A square
is a rhombus whose angles
are each 90°.
Try it!
Did You Know?
A quadrilateral can also be called
a quadrangle. The meaning is
still the same: a figure with four
angles and four straight sides.
Many colleges have quadrangles
at the center of their campuses.
This is Mob Quad at Merton
College, Oxford, England.
Chapter 3 — Geometry
210
Methodologies
Using Geometric Formulas
►
►
Example 1: Find the area of a parallelogram that has a base of 2 feet and a height of 18 inches.
Example 2: Find the area of a square with sides of 3 yards.
Steps in the Methodology
Step 1
Draw or
examine a
sketch of the
information
Step 2
Determine
which
formula(s) to
use
Step 3
Determine the
units needed
Example 1
Make a sketch if
necessary.
18 in
2 ft
When writing the formula,
make sure that each
part is identified with
the information given.
Sometimes two or more
formulas will be needed to
complete the information.
A = bh
Once the formula is
chosen, look back to
determine what units are
required.
Area uses square units, so
square feet (ft2) or square
inches (in2) would be
appropriate units.
where
b = the base
h= the height
We choose to work in feet,
so our answer will be in
square feet.
Step 4
Make sure that
all units agree
Units must be the same.
Use common conversion
ratios to change units.
Change the units in the
diagram if necessary.
Units are given in feet and
inches.
Use a proportion
equation to convert
the units:
18 in 12 in
=
x ft
1 ft
18 in : 1 ft
x=
= 1.5 ft
12 in
Replace 18 in with 1.5 ft.
Try It!
Example 2
Section 3.4 — Quadrilaterals
211
Steps in the Methodology
Step 4 (con’t)
Make sure that
all units agree
Units must be the same.
Use common conversion
ratios to change units.
Change the units in the
diagram if necessary.
Example 1
Validate: Does 1.5 ft = 18 in?
18 in ? 12 in
=
1.5 ft 1 ft
18(1) =? 1.5(12)
18 = 18 �
1.5 ft
2 ft
Step 5
Substitute
given
measurements
into the
formula
Step 6
Solve
Find needed information
first. Round each
calculation to the desired
number of decimal places.
b = 2 ft, h = 1.5 ft
Make the calculation.
Units multiply like
numbers.
A = (2 ft)(1.5 ft)
A = (2 ft)(1.5 ft)
A = (2 × 1.5) ft2
A = 3 ft2
Step 7
Validate:
• compare
units
• check
computations
Two steps: first compare
calculated units to the
anticipated units, then
validate calculations.
A good way to validate
your calculations is to
substitute your solution
back into the formula and
solve for one of the given
values (or for the single
given value, if there are
only two variables in the
formula).
• ft2 was anticipated 
• Using the calculated
area and the given base,
solve for the height:
A = bh, 3 ft 2 = 2 ft (h)
h=
3 ft 2
= 1.5 ft 
2 ft
Example 2
Chapter 3 — Geometry
212
Models
Model 1: Perimeter
Find the perimeter of a rectangular garden 7 meters wide and 12.2 meters long.
Step 1
Step 5
7m
12.2 m
Step 2
P = 2L + 2W
Step 3
Perimeter is measured in linear
units. Answer will be in meters.
Step 4
All necessary information is given in
meters; units agree.
P = 2L + 2W
P = 2(12.2 m) + 2(7 m)
Step 6
P = 24.4 m + 14 m
Answer: P = 38.4 m
Step 7
• Answer in meters 
• P = 2L + 2W
38.4 = 2L + 2(7)
38.4 = 2L +14
24.4 = 2L, L = 12.2 m 
Model 2: Area
Find the area of a baseball infield measuring 90 feet between bases.
Step 1
90 ft
Step 5
A = (90 ft)2
Step 6
A = (90 ft)(90 ft)
Answer: Area = 8100 ft2
Step 2
A = s2
Step 3
square units (feet2)
Step 4
Necessary units are given in feet
Step 7
• square feet 
• A = s2
8100 = s 2
s = 8100 = 90 ft 
Section 3.4 — Quadrilaterals
213
Addressing Common Errors
Issue
Units do
not agree
Incorrect
Process
Find the area
of a rectangular
carpet runner
that is 12′ by
24″.
Correct
Process
Resolution
In geometric
formulas, units
must be the
same.
12′ must be changed to
inches or 24″ must be
changed to feet in order
for the units to agree.
12 ft = 144 in
24 in
A = 24 in × 144 in
= 3456 in2
OR
A = 12 × 24
= 288
12 ft
24 in
= 2 ft
A = 2 ft × 12 ft
= 24 ft2
Using the
wrong unit
How much
carpet is needed
for an area that
measures
12′ by 10′?
Round up to
the next whole
square yard.
A = 12 × 10
= 120 ft2
Determine
the units
requested in
the answer
before
beginning your
calculations.
The units are correct for
area, but are not the
requested units for the
problem (square yards).
Convert feet to yards
before proceeding.
Validation
• Area is square units;
units are inches so the
answer is square inches.

• 3456 in 2 = l × 144 in
3456 in 2
=l
144 in
l = 24 in 
OR
• square feet 
• 24 ft 2 = l × 12 ft
24 ft 2
=l
12 ft
l = 2 ft 
• square yards 
1
10
• 13 yd 2 '
yd =
3
3
40 2 10
=
yd '
yd
12 ft 3 ft
12
3
3
=
, x = = 4 yd
x yd 1 yd
3
 40 3 
=  ×  yd = 4 yd 
10 ft 3 ft
10
 3 10 
=
, x=
yd
x yd 1 yd
3
Note: > 13 yards of carpet
10
A = 4 yd ×
yd
is needed, so get 14
3
yards.
40 2
=
yd
3
1
= 13 yd 2
3
. 14 yd 2 rounded up to
the next whole
square yard
Chapter 3 — Geometry
214
Issue
Not
validating
units
Incorrect
Process
Find the area
of a field that
measures 42 feet
by 30 yards.
A = lw
Carrying
units along in
calculations
helps validate
that the work
was done
correctly.
A = 42 × 30
Validation
42 ft 3 ft
=
x yd 1 yd
42
x=
= 14 yd
3
A = lw
A = 14 yd × 30 yd
• Area is sq feet or sq
yards. The answer is in
square yards. 
• 420 yd 2 = l : 30 yd
420 yd 2
=l
30 yd
14 yd = l
14 yd 1 yd
=
x ft
3 ft
x = 42 ft 
= 420 yd 2
A = 1260 ft
2
Incorrect
drawing or
sketch
Correct
Process
Resolution
A parallelogram
has a base of 11
inches, a width
of 13 inches and
a height of 12
inches. What is
the perimeter?
12 in
11 in
The height
is the
perpendicular
distance from
the base to the
top of a figure.
12 in
13 in
11 in
Be sure to
check your
drawing
against the
information
provided.
P = 2b + 2w
Be sure to
verify what
shape you’re
working with
and that you
are applying
the correct
formula.
The quadrilateral is
identified in the problem
as a parallelogram (not a
rhombus).
• inches
48 in = 2
(11 in ) + 2 w
26 in = 2 w
•
w = 13 in 
= 2(11) + 2(13)
= 22 + 26
= 48 in
P = 2b + 2w
= 2(11) + 2(12)
= 22 + 24
= 46 in
Using an
incorrect
formula
Find the
perimeter of the
parallelogram
below:
8m
10 m
P = 4s
= 4(10)
= 40 m
The correct formula for
finding the perimeter of a
parallelogram is:
P = 2b + 2w
The correct calculation is:
P = 2(10) + 2(8)
= 36 m
• meters 
• 36 m = 2(10 m) + 2w
16 m = 2w
w=8m 
Section 3.4 — Quadrilaterals
215
Preparation Inventory
Before proceeding, you should be able to use the correct formulas to calculate the following:
Area and perimeter of a rectangle
Area and perimeter of a trapezoid
Area and perimeter of a parallelogram
Squares from a Parallelogram?
The squares in this drawing are all based
on the parallelogram. The top and bottom
squares each have sides the same length as
the bases of the parallelogram. The left and
right squares have sides the same length as
the sides of the parallelogram. When you
draw a line from the centers of each of
the squares you get a new square.
This particular idea is based on a problem
posed by French mathemetician Victor
Thébault. There are many more interesting
geometric problems based on quadrilaterals.
To learn more, try searching online.
Section 3.4
Activity
Quadrilaterals
Performance Criteria
• Finding the perimeter and area of quadrilaterals.
– use of the appropriate formula
– accuracy of calculations
– validation of the answer
Critical Thinking Questions
1. What are four applications for area?
2. Why is perimeter measured in linear units?
3. Why does area use square units?
4. Why do units have to be the same in order to find perimeter or area?
216
Section 3.4 — Quadrilaterals
217
5. What value does a sketch provide for solving a geometric problem?
6. Why is the height used in finding the area of parallelograms and trapezoids?
7. The formulas for finding the area of a rectangle and the area of a parallelogram are very similar. Why?
Tips
for
Success
• Good practice includes validating by correctly identifying units of measure: linear units for perimeter and
square units for area
• Draw and label a diagram or sketch as accurately as possible—use graph paper as a tool to help you
Chapter 3 — Geometry
218
Demonstrate Your Understanding
1. Find the perimeter as indicated for each of the following:
Problem
a)
a rectangle with
length 14 m and
width 27 m
b)
a rectangle with
length 3.5 feet and
width 28 inches
c)
Measurements of
the roof are:
top = 15 ft
bottom = 20 ft
sides = 10 ft
height = 8 ft
What is the length
of a string of lights
framing the front
of the roof (the
part visible in the
illustration)?
d)
A paper kite in a
rhombus shape has
sides of 30 in, a
height of 12 in. The
small diagonal is
15 in and the large
diagonal is 48 in.
How much fringe is
needed to go around
the kite?
Worked Solution
Validation
Section 3.4 — Quadrilaterals
219
2. Find the area as indicated for each of the following:
Problem
a)
a rectangle with
length 4 miles and
width 2.3 miles
b)
a rectangle with
length of 52 inches
and width of three
feet
c)
Measurements of
the roof are:
top = 20 ft
bottom = 34 ft
sides = 25 ft
height = 24 ft
Find the area of the
front of the roof (the
part visible in the
illustration).
Worked Solution
Validation
Chapter 3 — Geometry
220
Problem
d)
Worked Solution
Validation
A paper kite in a
rhombus shape has
sides of 30 in and a
height of 12 in. The
small diagonal
is 15 in and the
large diagonal is
48 in.
How much paper
was used to make
the kite? (Use the
diagram below as
needed.)
s
d1
s
h
d2
s
s
Section 3.4 — Quadrilaterals
Identify
and
221
Correct
the
Errors
In the second column, identify the error(s) in the worked solution or validate its answer.
If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer.
Worked Solution
1) A cabinet door measures
3 feet by 15 inches. What
is the area of the door?
3 ft
15 in
A = bh
= 3(15) = 45 inches
2) Find the area of a square
that measures 2.2 yards
on each side.
2.2 yd
Area = 4s
= 4 (2.2) = 8.8 yd
3) Find the area of a
parallelogram that has
adjacent sides of 3 feet
and 7 feet and a height of
2 feet.
3
2
7
A = bh
= 7 ft (2 ft) = 14 ft
Identify Errors
or Validate
Correct Process
Validation
Chapter 3 — Geometry
222
Identify Errors
or Validate
Worked Solution
4) Find the perimeter of the
trapezoid shown below:
30 in
29 in
38 in
2 ft
75 in
P = b1 + b2 + s3 + s4
= 75 in + 30 in + 29 in + 38 in
= 172 in
5) Find the area of a
rhombus-shaped kite if
each edge is 50 cm and
the height is 46 cm.
50
46
1
A = h (b1 + b 2)
2
1
= 46 cm (50 cm + 50 cm)
2
= 23 cm(100 cm)
= 2300 cm 2
Correct Process
Validation