Formula for the Area
of a Triangle
Objective To guide the development and use of a formula
ffor the area of a triangle.
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Common
Core State
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Ongoing Learning & Practice
Key Concepts and Skills
Solving Fraction Problems
• Find the areas of rectangles and
parallelograms. Math Journal 2, p. 243
Students identify fractional parts of
number lines, collections of objects,
and regions.
[Measurement and Reference Frames Goal 2]
• Develop a formula for calculating the area
of a triangle. [Measurement and Reference Frames Goal 2]
Math Boxes 8 7
• Identify perpendicular line segments and
right angles. [Geometry Goal 1]
Math Journal 2, p. 244
Students practice and maintain skills
through Math Box problems.
• Describe properties of and types of
triangles. [Geometry Goal 2]
Study Link 8 7
• Evaluate numeric expressions containing
parentheses. [Patterns, Functions, and Algebra Goal 3]
Math Masters, p. 266
Students practice and maintain skills
through Study Link activities.
Key Activities
Students arrange triangles to form
parallelograms. They develop and use a
formula for finding the area of a triangle.
Ongoing Assessment:
Recognizing Student Achievement
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
ENRICHMENT
Comparing Areas
Math Masters, p. 267
scissors
Students cut apart a regular hexagon and
use the pieces to make area comparisons.
ENRICHMENT
Finding the Area and Perimeter
of a Hexagon
Math Masters, p. 268
centimeter ruler
Students use a combination of different
area formulas to find the area of a
nonregular hexagon.
EXTRA PRACTICE
Playing Rugs and Fences
Math Masters, pp. 498–502
Student Reference Book, pp. 260 and 261
Students practice finding the perimeter and
area of polygons.
Use journal page 242. [Measurement
and Reference Frames Goal 2]
Key Vocabulary
equilateral triangle isosceles triangle scalene triangle right triangle base height
Materials
Math Journal 2, pp. 240 – 242
Study Link 8 6
Math Masters, pp. 265 and 454A
transparency of Math Masters, p. 403
(optional) slate centimeter ruler scissors tape index card or other
square-corner object
Advance Preparation
For the optional Extra Practice activity in Part 3, consider copying the Rugs and Fences Cards on Math Masters, pages 498–501 on cardstock.
Teacher’s Reference Manual, Grades 4–6 pp. 180–185, 221, 222
Lesson 8 7
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Getting Started
Mental Math and Reflexes
Math Message
Dictate large numbers for students to write on their slates.
Suggestions:
Make a list of everything that you know
about triangles.
1,234,895
60,020,597
365,798,421
367,891
500,602
695,003
3,020,300,004
6,000,000,500
90,086,351,007
For each number, ask questions such as the following:
• Which digit is in the millions place?
• What is the value of the digit x?
• How many hundred millions are there?
Study Link 8 6 Follow-Up
Have small groups compare answers and explain
their strategies for finding the length of the base
when the height and the area are given.
1 Teaching the Lesson
Math Message Follow-Up
NOTE Because isosceles triangles are
defined as having at least two sides of
equal length, all equilateral triangles are
isosceles. So, some students may include
the equilateral triangles in the isosceles
pile. If this occurs, you may want to have
the class decide on a definition of isosceles
triangle that would exclude the equilateral
triangles. Sample answer: An isosceles
triangle has exactly two sides that are the
same length.
As students share their responses, write them on the board.
The list might include:
A triangle is a three-sided polygon.
The sum of the measures of the angles in a triangle is 180°.
A triangle has three vertices.
A triangle is a convex polygon.
An equilateral triangle is a triangle in which all three sides
have the same measure, and all three angles have the same
measure. An equilateral triangle is a regular polygon.
Student Page
Date
LESSON
87
1.
Time
Areas of Triangles
An isosceles triangle is a triangle that has at least two sides
with the same length and at least two angles with the same
measure.
136
Cut out Triangles A and B from Math Masters, page 265.
DO NOT CUT OUT THE ONE BELOW. Tape the two triangles
together to form a parallelogram.
Triangle A
1 cm2
A scalene triangle is a triangle in which there are no sides of
equal length and no angles of equal measure.
Tape your parallelogram in the space below.
B
A
6
4
base =
height =
Area of triangle =
2.
6
4
base =
cm
height =
cm
12
cm2
A right triangle is a triangle with one 90° angle.
cm
Exploring Triangle Properties
cm
Area of parallelogram =
24
cm2
Tape your parallelogram in the space below.
Have students cut out the cards on Math Masters, page 454A. Ask
students to sort the triangles into three categories: equilateral,
isosceles, and scalene. equilateral: C, F; isosceles: E, G, H, N, P;
scalene: A, B, D, I, J, K, L, M, O.
D
C
base =
height =
4
4
Area of triangle =
base =
cm
height =
cm
8
cm2
4
4
cm
cm
Area of parallelogram =
16
Discuss the angle properties of each type of triangle: equilateral
triangles have three angles of equal measure, isosceles triangles
have two angles of equal measure, and scalene triangles have no
angles of equal measure.
cm2
Math Journal 2, p. 240
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WHOLE-CLASS
ACTIVITY
(Math Masters, p. 454A)
Do the same with Triangles C and D on Math Masters, page 265.
Triangle C
WHOLE-CLASS
ACTIVITY
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Draw a right triangle on the board:
Ask students to tell you the definition of a right triangle. A right
triangle is a triangle with one 90° angle.
Ask: Do you see any right triangles in any of the piles? Yes.
Triangles A, H, I, and P are right triangles. Can an equilateral
triangle be a right triangle? Explain. No. All the angles in an
equilateral triangle have the same measure. Can an isosceles
triangle be a right triangle? Explain. Yes. Triangles H and P are
isosceles and right. Can a scalene triangle be a right triangle?
Explain. Yes. Triangles A and I are scalene and right.
Right
Draw the chart shown in the margin on the board. Ask students to
help you fill in the chart with “yes” and “no” to show which types of
triangles can be right triangles.
Equilateral
no
Isosceles
yes
Scalene
yes
Tell students that in this lesson they will develop a formula for the
area of a triangle using the formula for the area of a parallelogram.
height
Links to the Future
base
The use of a formula to calculate the area of a triangle is a Grade 5 Goal.
Developing a Formula for
the Area of a Triangle
The height of a triangle is measured
along a line segment perpendicular
to the base. As with parallelograms,
any side of a triangle can be the base.
The choice of the base determines
the height.
WHOLE-CLASS
ACTIVITY
PROBLEM
PRO
P
RO
R
OB
BLE
BL
L
LE
LEM
EM
SO
S
SOLVING
OL
O
LV
L
VIN
V
IIN
NG
(Math Journal 2, pp. 240 and 241;
Math Masters, p. 265)
Student Page
Date
LESSON
87
3.
Triangle E
base =
height =
Ask the class to turn to journal page 240 while you distribute
copies of Math Masters, page 265. Point out that Triangles A and
B on the master are the same as Triangle A on the journal page.
Guide students through the following activity:
3. Tape the parallelogram in the space next to Triangle A in
the journal.
7
2
base =
cm
height =
cm
7
Area of triangle =
4.
2. Tape the triangles together at the shaded corners to form a
parallelogram.
Tape your parallelogram in the space below.
E
The shortest distance from the vertex above the base to the base is
called the height of the triangle. Draw a dashed line to show the
height and label it. Include a right-angle symbol. (See margin.)
1. Cut out Triangles A and B from the master. Make sure students
realize that the triangles have the same area and are congruent.
continued
Do the same with Triangles E and F.
F
Draw a triangle on the board. Choose one of the sides—the side on
which the triangle “sits,” for example—and call it the base. Label
the base in your drawing. Explain that base is also used to mean
the length of the base.
Time
Areas of Triangles
cm2
7
2
cm
cm
Area of parallelogram =
14
Do the same with Triangles G and H.
Triangle G
Tape your parallelogram in the space below.
H
G
base =
height =
4
3
Area of triangle =
5.
base =
cm
height =
cm
6
cm2
4
3
cm
cm
Area of parallelogram =
12
cm2
Write a formula for the area of a triangle.
(b ∗ h)
A = _12 ∗ (b ∗ h), or A = _
2
height
length of base
Math Journal 2, p. 241
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Student Page
Date
Time
LESSON
Areas of Triangles
87
6.
Draw a line segment to show the height of Triangle SAM.
Use your ruler to measure the base and height of the
triangle. Then find the area.
5
2
2
base =
height =
Area =
7.
Discuss the relationship between the area of the triangle and the
area of the parallelogram. Triangles A and B have the same area.
Therefore, the area of either triangle is half the area of the
parallelogram.
136
continued
S
cm
cm
A
M
4. Record the dimensions and areas of the triangle and the
parallelogram. Base of triangle and parallelogram = 6 cm;
height of triangle and parallelogram = 4 cm; area of
parallelogram = 24 cm2; area of triangle = _12 the area of
parallelogram = 12 cm2.
cm2
Draw three different triangles on the grid below. Each triangle must have
an area of 3 square centimeters. One triangle should have a right angle.
Sample answers:
8.
Have students repeat these steps with Triangles C and D, E and
F, and G and H. Then bring the class together to state a rule and
write a formula for the area of a triangle.
See the shapes below. Which has the larger area—the star or the square? Explain your answer.
Neither. Both have an area of 16 sq units. Area of
square = 4 ∗ 4 = 16; area of star = area of square in
center + (4 ∗ area of a triangle) = 4 + (4 ∗ 3) = 16.
Since the base and the height of a triangle are the same as the
base and the height of the corresponding parallelogram, then:
Area of the triangle = _12 the Area of the parallelogram, or
Area of the triangle = _12 of (base ∗ height)
Math Journal 2, p. 242
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Using variables:
A = _12 of (b ∗ h), or A = _12 ∗ (b ∗ h)
where b is the length of the base and h is the height.
Have students record the formula at the bottom of journal
page 241.
Solving Area Problems
PARTNER
ACTIVITY
(Math Journal 2, p. 242; Math Masters, p. 403)
Algebraic Thinking Work with the class on Problem 6. Students
can place an index card (or other square-corner object) on top of
the triangle, align the bottom edge of the card with the base
(making sure that one edge of the card passes through point S),
and then draw a line for the height. Students will need a
centimeter ruler to measure the base and the height.
Student Page
Date
Time
LESSON
Fractions of Sets and Wholes
87
1.
Circle _6 of the triangles. Mark Xs on
_2 of the triangles.
3
1
2. a.
b.
59
2
Shade _5 of the pentagon.
index card
S
height
3
Shade _5 of the pentagon.
A
3.
5.
a.
How many musicians
play the flute?
14
b.
How many musicians
play the trombone?
7
of
120
_5
c. 6 of 120 is
_1
3
e.
Wei had 48 bean-bag animals in her
collection. She sold 18 of them to another
collector. What fraction of her collection
did she sell?
_3
8
Complete.
_3
a. 4
6.
4.
There are 56 musicians in the school band:
_1 of the musicians play the flute and _1 play
4
8
the trombone.
100
3
3
_
d. 10 of
.
_5
f. 4
of 72 is 24.
of 27 is 18.
50
of 16 is
Adjusting the Activity
ELL
On the board, draw a triangle that has an obtuse angle as one of its
base angles. Have students draw the height of the triangle. Demonstrate by
extending the base along the side of the obtuse angle and drawing a
perpendicular line from the opposite vertex to the extended base. For example:
_2
b.
is 90.
M
is 15.
20
.
Fill in the missing fractions on the number line.
0
1
6
2
6
3
6
4
6
5
6
1
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
Math Journal 2, p. 243
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Student Page
Date
Time
LESSON
Math Boxes
87
It may surprise some students that the star and the square
in Problem 8 have the same area. One way to find the area of
the star is to think of it as a square with a triangle attached to
each of its sides. (See margin.)
1.
2.
Write three equivalent fractions for
each fraction.
8
_
_4
a. 9
18
6
_
16
4
_
10
14
_
_3
b. 8
_2
c. 5
7
_
d. 10
20
Sample answers:
20
_
45
,
,
15
_
40
10
_
25
35
_
,
,
,
,
50
,
,
Measure the sides of the figure to the
nearest centimeter to find its perimeter.
40
_
90
30
_
80
20
_
cm
There are many possibilities for Problem 7. You can use a
transparency of a 1-cm grid (Math Masters, page 403) on the
overhead projector to display a number of them.
2
Have students complete Problems 7 and 8.
2
2
50
70
_
100
6
Complete the “What’s My Rule?” table,
and state the rule.
Rule:
4.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 242, Problem 8 to assess students’ ability to describe a
strategy for finding and comparing the areas of a square and a polygon.
Students are making adequate progress if they are able to count unit squares
and partial squares to find the areas of these two shapes. Some students may
describe the use of a formula to calculate the areas of the triangles.
[Measurement and Reference Frames Goal 2]
5.
in
out
A
70 times
8.69
4.09
B
100 times
11.03
6.43
C
50 times
19.94
15.34
D
210 times
26.05
21.45
b.
Divide with a paper-and-pencil algorithm.
7,653 6 =
is half as much as 86.
24
48 is twice as much as
c.
150
d.
1
40 is _5 of
e.
135
131
81
6.
43
cm
cm
162–166
Complete.
a.
cm
If you throw a die 420 times, about how
many times would you expect
to
come up? Circle the best answer.
-4.6
Journal
page 242
Problem 8
17
Perimeter =
49–51
3.
5
cm
cm
1,275 _36 , or 1,275 _12
.
is 3 times as much as 50.
200
.
is 5 times as much as 27.
22 23
179
Math Journal 2, p. 244
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2 Ongoing Learning & Practice
Solving Fraction Problems
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 243)
Students identify fractional parts of number lines, collections of
objects, and regions.
Math Boxes 8 7
INDEPENDENT
ACTIVITY
Study Link Master
Name
Date
STUDY LINK
(Math Journal 2, p. 244)
Time
Areas of Triangles
87
136
Find the area of each triangle.
2.
4'
8'
Number model:
16
Area =
12 cm
_1 ∗ (8 ∗ 4) = 16
2
square feet
_1
Number model: 2
30
Area =
3.
∗ (12 ∗ 5) = 30
square cm
4.
2 in.
34 cm
Writing/Reasoning Have students write a response to the
following: For Problem 4, write two probability questions
for which the correct answer would be D—210 times.
Sample answers: If you throw a die 420 times, about how many
times would you expect an even number to come up? If you throw a
die 1,260 times, how many times would you expect a 6 to come up?
1.
5 cm
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 8-5. The skill in Problem 6
previews Unit 9 content.
75 cm
10 in.
_1
Number model: 2
10
Area =
∗ (10 ∗ 2) = 10
square in.
_1
Number model: 2
Area =
∗ (34 ∗ 75) = 1,275
1,275 square cm
Try This
The area of each triangle is given. Find the length of the base.
Study Link 8 7
base =
INDEPENDENT
ACTIVITY
3
in.
12 in.
(Math Masters, p. 266)
Home Connection Students calculate the areas of
triangles. They continue to work on Math Masters,
page 262, which should be completed before Lesson 9-1.
6.
Area = 18 in2
Area = 15 m2
6
base =
5m
5.
m
?
?
Practice
7.
18,
27 , 36 , 45, 54 , 63, 72
8.
8
, 16,
24 , 32, 40 , 48 , 56
Math Masters, p. 266
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Lesson 8 7
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Teaching Master
Name
Date
LESSON
87
䉬
Time
1.
Cut out the hexagon below. Then cut out the large equilateral triangle.
You should end up with one large triangle and three smaller triangles.
2.
Use the large triangle and the three smaller triangles to form a rhombus.
3.
3 Differentiation Options
Comparing Areas
PARTNER
ACTIVITY
ENRICHMENT
a.
Sketch the rhombus in the
space to the right.
b.
Is the area of the rhombus the
same as the area of the hexagon?
c.
Is it possible for two different
shapes to have the same area?
Comparing Areas
yes
5–15 Min
(Math Masters, p. 267)
yes
Put all the pieces back together to form a hexagon with an equilateral
triangle inside.
How can you show that the area of the hexagon is twice the area of the
large triangle?
Sample answer: The three smaller triangles
cover the equilateral triangle. Six of the
smaller triangles cover the entire hexagon.
To apply students’ understanding of area, have them compare the
areas of a rhombus and a hexagon.
INDEPENDENT
ACTIVITY
ENRICHMENT
Finding the Area and
There are 4 triangles in the hexagon.
䉬 The large triangle is called an equilateral
triangle. All 3 sides are the same length.
䉬 The smaller triangles are called isosceles
triangles. Each of these triangles has
2 sides that are the same length.
5–15 Min
Perimeter of a Hexagon
(Math Masters, p. 268)
To apply students’ understanding of area formulas, have
them find the area and perimeter of a nonregular
hexagon. Counting squares to find the area is not
permitted; students are encouraged to divide the hexagon into
figures and then use a formula to calculate the area of each figure.
Math Masters, p. 267
One strategy is to partition the polygon as shown below.
Another strategy can be found on the reduction of Math Masters,
page 268.
Teaching Master
Name
LESSON
8 7
䉬
1.
Date
B
Area and Perimeter
Find the area of the hexagon below without counting squares.
Hint: Divide the hexagon into figures for which you can calculate
the areas: rectangles, parallelograms, and triangles. Use a formula
to find the area of each of the figures. Record your work.
Total area of hexagon ⫽
2.
A
Time
126
C
131
134–136
cm2
Find the perimeter of the hexagon. Use a centimeter ruler.
Perimeter ⫽
48
cm
Sample answer:
10 cm
24 cm2
EXTRA PRACTICE
12 cm
5 cm
48 cm2
6 cm2
2
8 cm 24 cm
24 cm2
10 cm
Playing Rugs and Fences
PARTNER
ACTIVITY
15–30 Min
(Math Masters, pp. 498–502; Student Reference Book,
pp. 260 and 261)
To practice calculating the area and perimeter of a polygon, have
students play Rugs and Fences. See Lesson 9-2 for additional
information.
1 cm
3 cm
Math Masters, p. 268
698
Unit 8 Perimeter and Area
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Name
Date
Time
Exploring Triangle Properties
B
C
A
D
H
G
F
E
I
Copyright © Wright Group/McGraw-Hill
J
M
K
L
P
N
O
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