Area 8.1 Rectangles & Parallelograms 8.2 Triangles, Trapezoids,& Kites.

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