area 8.1 rectangles & parallelograms 8.2 triangles, trapezoids,& kites
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Area
8.1 Rectangles & Parallelograms8.2 Triangles, Trapezoids ,& Kites
Rectangles & Parallelograms
•A= bh,
•where b is the length of the base and h is the height of the rectangle.
Height, h = 3 units
Base, b= 6 units
6
3
A = 6 3 = 18 sq. units
Rectangles & Parallelograms
•, A= bh,
• where b is the length of the base and h is the height of the parallelogram.
Height, h = 3 units
Base, b= 6 units
6
3
A = 6 3 = 18 sq. units
AREA OF A PARALLELOGRAM
Discover the formula for area of a parallelogram.
h
b
AREA OF A PARALLELOGRAM
To do this let’s cut the left triangle and…
h
b
slide it…
AREA OF A PARALLELOGRAM
h
h b
slide it…
AREA OF A PARALLELOGRAM
h
h
b
slide it…
AREA OF A PARALLELOGRAM
h
h
b
slide it…
AREA OF A PARALLELOGRAM
h
hb
…thus, changing it to a rectangle.
What is the area of this rectangle?
AREA OF PARALLELOGRAM
h
b
bhA
AREA OF A PARALLELOGRAM
Since the area of the rectangle and parallelogram are the same, just rearranged, what is the formula for the area of this parallelogram?
h
b
bhA
A Closer Look at Parallelograms
Height
Length
Side
leng
th
Note that the height of the parallelogram is an altitude, perpendicular to the bases, and that it is used in determining the area and not the side lengths.
Practice Problems for 8.112 cm
6 cm
4 cm
6 cm
What is the area of the shaded region?
A =( 12 x 6) – (4x 6) = 72 – 24 =
A = 48 cm 2
5 cm
8 cm
9.4 cm What is the area of the shaded region?
A = ½ ( 8 x 5) = 20 cm 2
Homework 8.1 8.1/1-13,17,20
Triangles, Trapezoids, & Kites
h
Base, b
h
b1
b2
AREA OF A TRIANGLE
Discovering the formula for area of a
triangle.h
b
AREA OF A TRIANGLE
divide the triangle so that the height is divided in two
equal parts
b
½ h
½ h
AREA OF A TRIANGLE
Now take the top and rotate…
b
Remember, we divided the height
into two equal parts.
½ h
½ h
AREA OF A TRIANGLE
rotate…
½ h
½ h
b
AREA OF A TRIANGLE
b
rotate…
½ h
½ h
AREA OF A TRIANGLE
b
rotate…
½ h
½ h
AREA OF A TRIANGLE
b
rotate…½ h
½ h
AREA OF A TRIANGLE
b
rotate…½ h
½ h
AREA OF A TRIANGLE
b…until you have a parallelogram.
How would you represent the height of this parallelogram?
h21
½ h
½ h
AREA OF A TRIANGLE
b
b
Remember, you divided the height in two.
½ h
½ h½ h
½ h
AREA OF A TRIANGLE
½ h
b
What is the area of this parallelogram?
hbA 21
AREA OF A TRIANGLE
The formula for the area of a triangle is…
what?
h
b
221 hbhbA
AREA OF A TRAPEZOID
Let’s derive the formula for the area for a trapezoid.
AREA OF A TRAPEZOID
Remember, there are two different bases on a trapezoid.
1b
2b
h
AREA OF A TRAPEZOID
1b
2b
½ h
First divide the trapezoid horizontally so the height is divided in two equal parts.
½ h
AREA OF A TRAPEZOID1b
2b
Remember, we divided the height in two. Now, rotate…
½ h
½ h
AREA OF A TRAPEZOID
1b
2b
rotate…
½ h
½ h
AREA OF A TRAPEZOID
1b2b
…until, you have a parallelogram.
½ h ½ h
AREA OF A TRAPEZOID
How would you represent the height of this parallelogram?
h21
1b2b
½ h ½ h
AREA OF A TRAPEZOID
1b
2b
Remember, we divided the height in two.
½ h
½ h 1b2b
½ h ½ h
AREA OF A TRAPEZOID
How would you represent the base of this parallelogram?
)( 12 bb
1b2b
½ h ½ h
AREA OF A TRAPEZOID
1b
2b
The new base is made by connecting the top and bottom bases.
1b2b
½ h ½ h
½ h
½ h
AREA OF A TRAPEZOID
How would you represent the area of this parallelogram?
2)(
2121 21)( bbhbbhA
1b2b
½ h ½ h
AREA OF A TRAPEZOID
The area of this trapezoid is the same as the parallelogram.
What is the formula for area of a trapezoid?
1b
2b
h
2)(
2121 21)( bbhbbhA
Triangles, Trapezoids,& Kites
d2
d1
A1 = ½h1d1 ; A2 = ½h2d1
A = ½h1d1 + ½h2d1 h1
h2A = ½ d1(h1 + h2)
Since d2 = (h1 + h2)
A = ½ d1 d2
Triangles, Trapezoids, & Kites
h
Base, b
A =½ bh
h
b1
b2
A = ½ h (b1 + b2)
Practice Problems for 8.2
h = 4 cm
b = 11 cm
A= ½bh; A = ½(11)(4)
A = ½(44) = 22 cm2
h = 6 cm
b2 = 15 cm
b1 = 8 cm
A = ½h(b1 + b2)
A = ½(6)(8 + 15)
A = (3)(23) = 69 cm2
Homework 8.2 8.2/1-12, 20,25-28
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