Area of Trapezoids and Irregular Figures
Lesson
Area
[
OBJECTIVE
]
•
The student will find the area of trapezoids
and irregular figures by composing or
decomposing them into triangles and
rectangles in the context of mathematical and
real-world problems.
[
MY
SKILLS
]
•
Area of rectangles
[
ESSENTIAL
QUESTIONS
]
1.
Explain how to use a rectangle to find the
area of a triangle.
2.
Explain how to use the area of a triangle and
rectangle to find the area of a trapezoid.
3.
Explain how to find the area of irregular
shapes.
[Warm-Up]
Begin by completing the warm-up for this
lesson.
AREA
SOLVE Problem – Introduction
[
LESSON
]
SOLVE
Marlina and her friend, Anise, are working on a
drawing in art class. They have been given a piece
of rectangular construction paper and asked to
share it by dividing it into two right triangles. The
length of the base of the paper is 18 inches, and
the height of the paper is 9 inches. What is the
area of one triangle?
[
LESSON
]
SOLVE
S
Study the Problem
Underline the question.
[
LESSON
]
SOLVE
Marlina and her friend, Anise, are working on a
drawing in art class. They have been given a piece
of rectangular construction paper and asked to
share it by dividing it into two right triangles. The
length of the base of the paper is 18 inches, and
the height of the paper is 9 inches. What is the
area of one triangle?
[
LESSON
]
SOLVE
S
Study the Problem
Underline the question.
This problem is asking me to find
the area of one triangle.
REVIEW OF AREA OF RECTANGLES
Review of Area of Rectangles
Discuss and share ideas of real-life situations where
we may need to determine the area.
•
Room dimensions for carpet
•
Paint for walls
•
Grass seed for a yard
Review of Area of Rectangles
What is the shape of the picture in the first box?
Rectangle
What is one strategy we can use to determine the
area of the rectangle?
We can count the number of unit squares in the figure.
Review of Area of Rectangles
Is there another way we can find the area?
Yes, we can use the formula of
Area = length times width
What is the length of the rectangle?
6 units
6 units
Review of Area of Rectangles
What is the width of the rectangle?
3 units
What is the formula we use to find the area?
A
=
lw
6 units
3 units
A
=
lw
Review of Area of Rectangles
Substitute the values for our formula.
A
=
lw
A
= (6)(3)
What is the area of the rectangle?
A
= 18 units
2
6 units
3 units
A
=
lw
A
= 6(3)
A
= 18 units
2
Review of Area of Rectangles
How is this picture different from the sample problem?
The shape is not divided into cubes, but the length and
width are given.
Do we have the information we need to find the
area?
Yes
5 in.
4 in.
Review of Area of Rectangles
What is the length of the rectangle?
5 inches
What is the width of the rectangle?
4 inches
5 in.
4 in.
5 inches
4 inches
Review of Area of Rectangles
What is the area of the rectangle?
A
=
lw
A
= (5)(4)
5 in.
4 in.
5 inches
4 inches
A
=
lw
A
= 5(4)
A
= 20 in.
2
A
= 20 in.
2
Review of Area of Rectangles
How is this problem different from Problem 1?
The length and width of the rectangle are given
without a picture.
Draw a representation of the rectangle in the first
column and label the dimensions.
12 in.
4 in.
12 in.
4 in.
Review of Area of Rectangles
What is the formula we use to find the area?
A
=
lw
What is the area of the rectangle?
12 in.
4 in.
12 in.
4 in.
A
=
lw
A
= 12(4)
A
= 48 in.
2
A
= (12)(4)
A
= 48 in.
2
AREA OF TRIANGLES
Area of Triangles
Use a ruler to create a rectangle with a length of 8 units
and a width of 5 units.
Area of Triangles
Explain how you can determine the area of the
rectangle.
Count the unit squares inside the shape or multiply
the length times the width.
Area of Triangles
What is the area of the rectangle?
Length × Width
8 × 5
40 square units
Area of Triangles
Cut the rectangle into two triangles.
Now, use a ruler to draw a line from the lower
left-hand corner to the upper right-hand corner.
Area of Triangles
Place the triangles back together to form the
rectangle.
We will use the area of a rectangle to discover the area
of a triangle.
Area of Triangles
40 units
2
What is the area of the rectangle?
Area of Triangles
They are the same size, or congruent.
What do you notice about the two triangles when you
place one on top of the other?
If the two congruent triangles combine to make the
rectangle, what is the relationship between the area of
one of the triangles and the rectangle?
Area of Triangles
The area of one triangle is half of the rectangle.
What was the area of the rectangle?
Area of Triangles
40 square units
If half of the rectangle is the area of one triangle, what
is the area of the triangle?
20 square units
Place the triangle on your work area with the right
angle of the triangle on the right
Area of Triangles
Record the area of the triangle.
A
= 20 units
2
What is the formula for the area of the rectangle?
Area of Triangles
The area of a rectangle is found by multiplying the
length times the width.
A
=
lw
Area of Triangles
What is another term we can use to describe the width
of the rectangle?
height
What is another term we can use to describe the
length of the rectangle?
The base
Area of Triangles
What is the height of the triangle?
5 units
What is the length of the base of the triangle?
8 units
8 units
5 units
Area of Triangles
Explain your thinking about this.
Two congruent triangles are equal to the rectangle so
one of the triangles is one half of the rectangle.
How can we describe the triangle in relation to the
rectangle using fractions?
The triangle is one half the size of the rectangle
8 units
5 units
Area of Triangles
How can we use the information we have to determine
the area of the triangle.
Use the area of the rectangle and divide by two or
multiply by one half.
What fractional part of the whole rectangle does the
triangle represent?
8 units
5 units
Area of Triangles
What is the length of the base of the triangle?
10 ft
What is the height of the triangle?
10 ft
8 ft
10 ft
8 ft
8 ft
Area of Triangles
What is the formula we discovered for finding the area
of a triangle?
10 ft
8 ft
10 ft
8 ft
Area of Triangles
Substitute the values and solve for the area.
10 ft
8 ft
10 ft
8 ft
Area of Triangles
What is the area of the triangle?
The area is 40 ft
2
.
10 ft
8 ft
10 ft
8 ft
FINDING AREA OF TRAPEZOIDS
Finding Area of Trapezoids
We can use the information that we know about the
area of rectangles and triangles to find the area of
trapezoids and other special polygons.
What figure is pictured above?
Trapezoid
12 cm
6 cm
4 cm
5 cm
Finding Area of Trapezoids
Do we know a formula for the area of a trapezoid?
No
What information do we know?
Finding Area of Trapezoids
Explain the relationship between the two bases of the
figure.
They are parallel.
Explain how you know they are parallel.
There are right angles drawn on the figure which
means that the two bases of the trapezoid are
perpendicular to the height connecting them.
Finding Area of Trapezoids
What is the measure of the longer base?
12 cm
What is the measure of the shorter base?
6 cm
Finding Area of Trapezoids
What is the measure of the slanted side?
5 cm
What is the perpendicular height of the trapezoid?
12 cm
6 cm
4 cm
5 cm
4 cm
Finding Area of Trapezoids
Is there a strategy we can use to find the area of the
trapezoid?
We can divide the shape into two triangles and one
rectangle.
12 cm
6 cm
4 cm
5 cm
Finding Area of Trapezoids
Based on the information given, can we find the base of
the triangle?
Yes
12 cm
6 cm
4 cm
5 cm
Explain your answer.
We know that both bases of the rectangle are 6 cm
because they must be congruent.
6 cm
Finding Area of Trapezoids
What do the tick marks mean on the figure?
The bases of the two triangles are congruent.
How can we use the information given to find the base
of the triangle? Why?
Subtract 6 from 12 and then divide the difference by 2.
12 – 6 = 6 ÷ 2 = 3
6 cm
Finding Area of Trapezoids
What is the base of each triangle?
3 cm
6 cm
3 cm
3 cm
Finding Area of Trapezoids
What is the perpendicular height of each triangle?
4 cm
6 cm
6 cm
4 cm
5 cm
3 cm
3 cm
Explain.
They are congruent.
4 cm
Finding Area of Trapezoids
Do we now have enough information to determine the
area of this figure?
Yes
6 cm
6 cm
4 cm
5 cm
3 cm
3 cm
What plan can we use to find the area of the trapezoid?
Find the area of each triangle and add those areas to
the area of the rectangle.
4 cm
Finding Area of Trapezoids
Now we can use our plan to write a formula and
substitute the information from the drawing to
determine the area.
6 cm
6 cm
4 cm
5 cm
3 cm
3 cm
4 cm
Finding Area of Trapezoids
6 cm
6 cm
4 cm
5 cm
3 cm
3 cm
4 cm
Area = 6 + 6 + 24
Area = 36 cm
2
Finding Area of Trapezoids
What is the area of the trapezoid?
36 cm
2
6 cm
6 cm
4 cm
5 cm
3 cm
3 cm
4 cm
PRACTICE WITH AREA
Practice with Area
Sometimes we have shapes that are irregular. We don’t
know a formula for these unusual shapes, but we can
use what we know about area of other shapes like
rectangles and triangles to find the total area. When a
shape is composed of more than one shape, we call it a
________________.
composite shape
These shapes can be taken apart or ____________ into
other shapes. Let’s look at the drawing of Carson’s
backyard.
decomposed
Practice with Area
What is the shape of the figure?
It has 6 sides so it is a hexagon.
Practice with Area
Do you know a formula for the area of a hexagon?
No
Practice with Area
What strategy can we use to find the area of a hexagon?
Divide the shape into two rectangles.
6 feet
10 feet
22 feet
38 feet
50 feet
40 feet
Vegetable
Garden
Practice with Area
Let’s sketch Shape 1 and Shape 2.
6 feet
10 feet
22 feet
38 feet
50 feet
40 feet
Vegetable
Garden
Practice with Area
Explain how to find the area of Shape 1.
50 feet
40 feet
Use the formula for area of a rectangle and
substitute the values from the shape.
A
=
lw
A
= 50 • 40
A
= 2,000 ft
2
Shape 1
Practice with Area
6 feet
10 feet
22 feet
38 feet
Vegetable
Garden
A
=
lw
A
= 16 • 60
A
= 960 ft
2
Explain how to find the area of Shape 2.
Use the formula for area of a rectangle and
substitute the values from the shape.
Shape 2
Practice with Area
What is the total area of the yard?
2000 + 960 = 2,960 ft
2
Shape 2
A
= 960 ft
2
Shape 1
A
= 2,000 ft
2
Practice with Area
What is a strategy we can use to determine the area
without the vegetable garden?
Find the area of the garden and subtract it from the
total area.
6 feet
10 feet
22 feet
38 feet
50 feet
40 feet
Vegetable
Garden
Practice with Area
6 feet
10 feet
22 feet
38 feet
50 feet
40 feet
Vegetable
Garden
A
= 110 ft
2
A
= 2,960 – 110 = 2,850 ft
2
Practice with Area
What is the area of the yard
without the vegetable garden?
6 feet
10 feet
22 feet
38 feet
50 feet
40 feet
Vegetable
Garden
2,850 ft
2
Now look at the figure on the next
page and see if you can decompose
the figure to find the area.
FOLDABLE
Foldable
Fold one corner of the piece
of paper down to the edge of
the other side of the paper.
Cut off the strip at the
bottom. A square should be
left.
Foldable
Cut off the bottom portion
that creates a long rectangle.
Cut off
Foldable
Unfold the paper back to its
original size to reveal a
square with a diagonal fold.
Foldable
Now, fold the top right
corner to match the bottom
left corner.
Foldable
After creasing the fold, open
the paper back to the
original square, now with
two diagonal folds.
Foldable
Next, fold each corner into
the center. Write “Triangle
Area” on one outside flap
and “Rectangle Area” on one
outside flap.
Fold all 4 corners
into the center.
AREA
SOLVE Problem – Completion
[
LESSON
]
SOLVE
Marlina and her friend, Anise, are working on a
drawing in art class. They have been given a piece
of rectangular construction paper and asked to
share it by dividing it into two right triangles. The
length of the base of the paper is 18 inches, and
the height of the paper is 9 inches. What is the
area of one triangle?
[
LESSON
]
SOLVE
S
Study the Problem
Underline the question.
This problem is asking me to find
the area of one triangle.
O
Organize the Facts
Identify the facts.
[
LESSON
]
SOLVE
Marlina and her friend, Anise, are working on a
drawing in art class. They have been given a piece
of rectangular construction paper and asked to
share it by dividing it into two right triangles. The
length of the base of the paper is 18 inches, and
the height of the paper is 9 inches. What is the
area of one triangle?
O
Organize the Facts
Identify the facts.
Eliminate the unnecessary facts.
[
LESSON
]
SOLVE
Marlina and her friend, Anise, are working on a
drawing in art class. They have been given a piece
of rectangular construction paper and asked to
share it by dividing it into two right triangles. The
length of the base of the paper is 18 inches, and
the height of the paper is 9 inches. What is the
area of one triangle?
O
Organize the Facts
Identify the facts.
Eliminate the unnecessary facts.
List the necessary facts.
•
Base length of rectangle: 18 inches
•
Height of rectangle: 9 inches
L
Line Up a Plan
Write in words what your plan of action will
be.
Put the values in the formula for the area of
a triangle for base and height.
Choose an operation or operations.
Multiplication
E
Examine Your Results
Does your answer make sense?
(Compare your answer to question.)
Yes, because we were looking for the area of
one triangle.
Is your answer reasonable?
(Compare your answer to the estimate.)
Yes, because it is close to our estimate of
about 80 inches
2
.
Is your answer accurate?
(Check your work.)
Yes
Write your answer in a complete sentence.
The area of one of the triangles is 81 inches
2
.
AREA
Closure
[
ESSENTIAL
QUESTIONS
]
1.
Explain how to use a rectangle to find
the area of a triangle.
If the area of a rectangle is divided into
two congruent triangles, the area of
each of the triangles is one half the
area of the rectangle.
[
ESSENTIAL
QUESTIONS
]
2.
Explain how to use the area of a triangle
and rectangle to find the area of a
trapezoid.
Divide the trapezoid into two triangles
and a rectangle. Find the area of each
figure and add them to find the total
area of the trapezoid.
[
ESSENTIAL
QUESTIONS
]
3.
Explain how to find the area of irregular
shapes.
Divide the irregular shape into figures
with known area formulas and add the
individual areas to find the total area of
the irregular shape.
Rectangle
Right Triangle
Area
Height
Base
Perpendicular
Congruent
Irregular Shapes
Composite
Compose
Decompose
Lesson
Area
How to find the area of trapezoids and irregular figures by decomposing them into triangles and rectangles in a mathematical and real-world context. Dive into essential questions, warm-up activities, and problem-solving scenarios to enhance your skills in this lesson.
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Presentation Transcript
Lesson Area
[OBJECTIVE] The student will find the area of trapezoids and irregular figures by composing or decomposing them into triangles and rectangles in the context of mathematical and real-world problems.
[MYSKILLS] Area of rectangles
[ESSENTIALQUESTIONS] 1. Explain how to use a rectangle to find the area of a triangle. 2. Explain how to use the area of a triangle and rectangle to find the area of a trapezoid. 3. Explain how to find the area of irregular shapes.
[Warm-Up] Begin by completing the warm-up for this lesson.
[LESSON] SOLVE Marlina and her friend, Anise, are working on a drawing in art class. They have been given a piece of rectangular construction paper and asked to share it by dividing it into two right triangles. The length of the base of the paper is 18 inches, and the height of the paper is 9 inches. What is the area of one triangle?
[LESSON] SOLVE S Study the Problem Underline the question.
[LESSON] SOLVE Marlina and her friend, Anise, are working on a drawing in art class. They have been given a piece of rectangular construction paper and asked to share it by dividing it into two right triangles. The length of the base of the paper is 18 inches, and the height of the paper is 9 inches. What is the area of one triangle?
[LESSON] SOLVE S Study the Problem Underline the question. This problem is asking me to find the area of one triangle.
Review of Area of Rectangles Discuss and share ideas of real-life situations where we may need to determine the area. Room dimensions for carpet Paint for walls Grass seed for a yard
Review of Area of Rectangles Picture Length Width Formula Area What is the shape of the picture in the first box? Rectangle What is one strategy we can use to determine the area of the rectangle? We can count the number of unit squares in the figure.
Review of Area of Rectangles Picture Length Width Formula Area 6 units Is there another way we can find the area? Yes, we can use the formula of Area = length times width What is the length of the rectangle? 6 units
Review of Area of Rectangles Picture Length Width Formula Area 6 units 3 units A = lw What is the width of the rectangle? 3 units What is the formula we use to find the area? A = lw
Review of Area of Rectangles Picture Length Width Formula Area A = 6(3) A = 18 units2 6 units 3 units A = lw Substitute the values for our formula. A = lw A = (6)(3) What is the area of the rectangle? A = 18 units2
Review of Area of Rectangles Picture Length Width Formula Area 1. 4 in. 5 in. How is this picture different from the sample problem? The shape is not divided into cubes, but the length and width are given. Do we have the information we need to find the area? Yes
Review of Area of Rectangles Picture Length Width Formula Area 1. 4 in.5 inches 4 inches 5 in. What is the length of the rectangle? 5 inches What is the width of the rectangle? 4 inches
Review of Area of Rectangles Picture Length Width Formula Area 1. A = 5(4) A = 20 in.2 4 in.5 inches 4 inches A = lw 5 in. What is the area of the rectangle? A = lw A = (5)(4) A = 20 in.2
Review of Area of Rectangles Picture Length Width Formula Area 2. 4 in. 4 in. 12 in. 12 in. How is this problem different from Problem 1? The length and width of the rectangle are given without a picture. Draw a representation of the rectangle in the first column and label the dimensions.
Review of Area of Rectangles Picture Length Width Formula Area 2. A = 12(4) A = 48 in.2 4 in. A = lw 4 in. 12 in. 12 in. What is the formula we use to find the area? A = lw What is the area of the rectangle? A = (12)(4) A = 48 in.2
Area of Triangles Use a ruler to create a rectangle with a length of 8 units and a width of 5 units.
Area of Triangles Explain how you can determine the area of the rectangle. Count the unit squares inside the shape or multiply the length times the width.
Area of Triangles What is the area of the rectangle? Length Width 8 5 40 square units
Area of Triangles Now, use a ruler to draw a line from the lower left-hand corner to the upper right-hand corner. Cut the rectangle into two triangles.
Area of Triangles We will use the area of a rectangle to discover the area of a triangle. Place the triangles back together to form the rectangle.
Area of Triangles What is the area of the rectangle? 40 units2
Area of Triangles What do you notice about the two triangles when you place one on top of the other? They are the same size, or congruent.
Area of Triangles If the two congruent triangles combine to make the rectangle, what is the relationship between the area of one of the triangles and the rectangle? The area of one triangle is half of the rectangle.
Area of Triangles What was the area of the rectangle? 40 square units If half of the rectangle is the area of one triangle, what is the area of the triangle? 20 square units
Area of Triangles A = 20 units2 Place the triangle on your work area with the right angle of the triangle on the right Record the area of the triangle.
Area of Triangles What is the formula for the area of the rectangle? The area of a rectangle is found by multiplying the length times the width. A = lw
Area of Triangles What is another term we can use to describe the length of the rectangle? The base What is another term we can use to describe the width of the rectangle? height
Area of Triangles 5 units 8 units What is the length of the base of the triangle? 8 units What is the height of the triangle? 5 units
Area of Triangles 5 units 8 units How can we describe the triangle in relation to the rectangle using fractions? The triangle is one half the size of the rectangle Explain your thinking about this. Two congruent triangles are equal to the rectangle so one of the triangles is one half of the rectangle.
Area of Triangles 5 units A = ? ?bh 8 units What fractional part of the whole rectangle does the triangle represent? ? ? How can we use the information we have to determine the area of the triangle. Use the area of the rectangle and divide by two or multiply by one half.
Area of Triangles Picture Length of Base Height Formula Area 2. 8 ft 10 ft 8 ft 10 ft What is the length of the base of the triangle? 10 ft What is the height of the triangle? 8 ft
Area of Triangles Picture Length of Base Height Formula Area A = ? 2. 8 ft 10 ft ?bh 8 ft 10 ft What is the formula we discovered for finding the area of a triangle? A = ? ?bh
Area of Triangles Picture Length of Base Height Formula Area A = ? A = ? A = 5(8) A = 40 ft2 2. 8 ft 10 ft ?bh ?(10)(8) 8 ft 10 ft Substitute the values and solve for the area. A = ? ?bh A = ? ?(10)(8) A = 5(8) A = 40 ft2
Area of Triangles Picture Length of Base Height Formula Area A = ? A = ? A = 5(8) A = 40 ft2 2. 8 ft 10 ft ?bh ?(10)(8) 8 ft 10 ft What is the area of the triangle? The area is 40 ft2.
Finding Area of Trapezoids 12 cm 4 cm 5 cm 6 cm We can use the information that we know about the area of rectangles and triangles to find the area of trapezoids and other special polygons. What figure is pictured above? Trapezoid
Finding Area of Trapezoids 12 cm 4 cm 5 cm 6 cm Do we know a formula for the area of a trapezoid? No What information do we know?
Finding Area of Trapezoids 12 cm 4 cm 5 cm 6 cm Explain the relationship between the two bases of the figure. They are parallel. Explain how you know they are parallel. There are right angles drawn on the figure which means that the two bases of the trapezoid are perpendicular to the height connecting them.
Finding Area of Trapezoids 12 cm 4 cm 5 cm 6 cm What is the measure of the longer base? 12 cm What is the measure of the shorter base? 6 cm
Finding Area of Trapezoids 12 cm 4 cm 5 cm 6 cm What is the measure of the slanted side? 5 cm What is the perpendicular height of the trapezoid? 4 cm
Finding Area of Trapezoids 12 cm 4 cm 5 cm 6 cm Is there a strategy we can use to find the area of the trapezoid? We can divide the shape into two triangles and one rectangle.
Finding Area of Trapezoids 12 cm 6 cm 4 cm 5 cm 6 cm Based on the information given, can we find the base of the triangle? Yes Explain your answer. We know that both bases of the rectangle are 6 cm because they must be congruent.
Finding Area of Trapezoids 6 cm 4 cm 5 cm 6 cm What do the tick marks mean on the figure? The bases of the two triangles are congruent. How can we use the information given to find the base of the triangle? Why? Subtract 6 from 12 and then divide the difference by 2. 12 6 = 6 2 = 3
