pre ctivity quadrilaterals preparation2+s 3+s 4 perimeter equals the sum of the lengths of each side...

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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.

• Find the perimeter of a quadrilateral• Find the area of a quadrilateral

Quadrilaterals

Section 3.4

new terms to LeArn

quadrilateral

Previously used

side

triangle

LeArning objectives

terminoLogy

�0� Chapter � — Geometry

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.

Following are profiles of five basic quadrilateral shapes, including their defining characteristics, perimeter, and area formulas, and observations or clarifying comments.

closed

not closed

A four-sided figure with opposite sides equal and

parallel. Each interior angle measures 90°

Rectangle

Perimeter (P) Area (A)P = 2l + 2w or

P = 2(l+w)

Perimeter equals twice the length plus

twice the width

A = lw

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.

w

l

l

w

A four-sided figure with opposite sides equal and

parallel. The interior angles are NOT necessarily equal

to 90°

Parallelogram

Perimeter (P) Area (A)P = 2b + 2w

Perimeter equals twice the length (base) plus

twice the width

A = bh

Area equals base times height

OBSERVATIONS: There are three important mea-surements for a parallelogram: base (length), side (width), and height (altitude). Height is required to find the area.

A

B

D

C

h

b

b

ww A four-sided figure with two unequal parallel sides and two non-parallel sides

Perimeter (P) Area (A)P = b1+b2+s3+s4

Perimeter equals the sum of the lengths of

each side

A=½h(b1+b2)

Area equals one-half times the height times the sum of the

bases

OBSERVATIONS: Many rooftops are trapezoidal in shape.

hs2

s1 (b1)

s4

s3 (b2) Trapezoid

A four-sided figure with all four sides equal. Each internal angle measures

90°

Perimeter (P) Area (A)

P = 4s

Perimeter equals four times the length of

one side

A = s2 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:

s

s

s

s

Square

�0�Sect�on �.� — Quadr�laterals

A four-sided figure with all sides equal. Its opposite

sides are parallel.

Rhombus

Perimeter (P) Area (A)P = 4s

Perimeter equals four times the length of

one side

A = sh

alternatively:

A = ½ d1d2

A

B C

D

d2 d1

OBSERVATIONS: A rhombus is sometimes called a diamond shape; think of a kite or a baseball infield.

A

B C

D

s

s

s

sh

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°.

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.

Try it!

Can you use the area formula for triangles to prove that:

given that the diagonals of a rhombus bisect (cut in half) each other at right angles?

Area d drhombus =12 1 2

��0 Chapter � — Geometry

Steps in the Methodology Example 1 Example 2

Step 1

Draw or examine a sketch of the information

Make a sketch if necessary.

18 in

2 ft

Step 2

Determine which formula(s) to use

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 whereb = the base h= the height

Step 3

Determine the units needed

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. 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 12

118 1

121 5

in ft

in ft

in ft in

ft

x

x

=

= =: .

Replace 18 in with 1.5 ft.

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.

►► Try It!

��Sect�on �.� — Quadr�laterals

Steps in the Methodology Example 1 Example 2

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.

Validate: Does 1.5 ft = 18 in?181 5

121

18 1 1 5 1218 18

in ft

in ft

.

( ) . ( )

=

== �

?

?

1.5 ft

2 ft

Step 5

Substitute given measurements into the formula

Find needed information first. Round each calculation to the desired number of decimal places.

b = 2 ft, h = 1.5 ft

A = (2 ft)(1.5 ft)

Step 6

Solve

Make the calculation. Units multiply like numbers.

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 ft (h)

h ft ft

ft

2

2

= =

= =32

1 5.

�� Chapter � — Geometry

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 7 • square feet •

A s8100 s

s ft

2

2

=

=

= =8100 90

Step 2 A = s2

Step 3 square units (feet2)

Step 4 Necessary units are given in feet

modeLs

Model 1: Perimeter

Find the perimeter of a rectangular garden 7 meters wide and 12.2 meters long.

Step 1 7 m

12.2 m

Step 5 P = 2L + 2WP = 2(12.2 m) + 2(7 m)

Step 6 P = 24.4 m + 14 mAnswer: P = 38.4 m

Step 7 • Answer in meters •

P L 2W38.4 2L 2(7)

LL L m

= +

= +

= +

= =

2

38 4 2 1424 4 2 12 2

.

. , .

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.


Addressing common errors

Issue Incorrect Process Resolution Correct

Process Validation

Units do not agree

Find the area of a rectangular carpet runner that is 12′ by 24″.

A = 12 × 24 = 288

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 in24 in

A = 24 in × 144 in = 3456 in2

OR12 ft

24 in = 2 ft

A = 2 ft × 12 ft = 24 ft2

• Area is square units; units are inches so the answer is square inches.

•

3456 1443456144

24

2

2

in in in in

in

=

=

=

l

l

l

×

OR

• square feet •

24 122412

2

2

2

ft ft ft ft

ft

=

=

=

l

l

l

×

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.

rounded up to the next whole square yard

12 31

123

4

10 31

103

ft yd

ft yd

yd

ft yd

ft yd

yd

xx

xx

= = =

= =

,

,

AA yd yd

yd

yd

yd

2

2

2

=

=

=

4 103

403

13 13

14

×

.

• square yards

• 2

2

1 1013 yd yd3 340 10yd yd3 340 3 yd 4 yd 3 10

=

=

= = ×

'

'

Note: > 13 yards of carpet is needed, so get 14 yards.


Issue Incorrect Process Resolution Correct

Process Validation

Not validating units

Find the area of a field that measures 42 feet by 30 yards.

A = lwA = 42 × 30A = 1260 ft2

Carrying units along in calculations helps validate that the work was done correctly.

42 31

423

14

14 30420

ft yd

ft yd

yd

A lwA yd yd

yd2

x

x

=

= =

=

=

=

×

• Area is sq feet or sq yards. The answer is in square yards.

•

420 3042030

1414 1

342

yd yd yd yd

yd yd ft

yd ft

2

2

=

=

=

=

=

l

l

l

xx

:

ft

Incorrect drawing or sketch

A parallelogram has a base of 11 inches, a width of 13 inches and a height of 12 inches. What is the perimeter?

11 in

12 in

P = 2b + 2w= 2(11) + 2(12)= 22 + 24= 46 in

The height is the perpendicular distance from the base to the top of a figure.

Be sure to check your drawing against the information provided.

11 in

13 in12 in

P = 2b + 2w= 2(11) + 2(13)= 22 + 26= 48 in

• inches

•

48 2 11 226 2

13

in in w in w

w in

= +

=

=

( )

Using an incorrect formula

Find the perimeter of the parallelogram below:

10 m

8 m

P = 4s= 4(10)= 40 m

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).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 = 2ww = 8 m


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.

��

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?

Section 3.4


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


demonstrAte your understAnding

Problem Worked Solution Validation

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 ftWhat 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?

1. Find the perimeter as indicated for each of the following:



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 ftFind the area of the front of the roof (the part visible in the illustration).

2. Find the area as indicated for each of the following:

��0 Chapter � — Geometry


d) 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.)

d1

d2s

s

s

h

s


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 Identify Errors or Validate Correct Process Validation

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.

2

7

3

A = bh= 7 ft (2 ft) = 14 ft

identify And correct the errors


Worked Solution Identify Errors or Validate Correct Process Validation

4) Find the perimeter of the trapezoid shown below:

75 in

38 in

30 in

29 in 2 ft

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

A h b b

46 cm 50 cm 50 cm

cm cm cm2

= +

= +

=

=

12

1223 1002300

1 2( )

( )

( )

pre ctivity quadrilaterals preparation2+s 3+s 4 perimeter equals the sum of the lengths of each side...

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