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Area of Parallelograms. Section 11.2. Goal. Find the area of parallelograms. Key Vocabulary. Base of a parallelogram Height of a parallelogram Parallelogram Rhombus . Parallelogram. A parallelogram is a quadrilateral where the opposite sides are congruent and parallel.

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Area of Parallelograms


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Presentation Transcript
slide2
Goal
  • Find the area of parallelograms.
key vocabulary
Key Vocabulary
  • Base of a parallelogram
  • Height of a parallelogram
  • Parallelogram
  • Rhombus
parallelogram
Parallelogram
  • A parallelogram is a quadrilateral where the opposite sides are congruent and parallel.
  • A rectangle is a type of parallelogram, but we often see parallelograms that are not rectangles (parallelograms without right angles).
the base of a parallelogram
The Base of a Parallelogram
  • Either pair of parallel sides of a parallelogram are called the bases of the parallelogram.

base

base

base

base

the height of a parallelogram
The Height of a Parallelogram
  • The shortest distance (perpendicular distance) between the bases of a parallelogram is called the height of the parallelogram.
  • The height of the parallelogram is always perpendicular to the bases.
area of parallelogram
Area of Parallelogram

What is the area

of this parallelogram?

h=height

b=base

area of parallelogram1
Area of Parallelogram

What is the area

of this parallelogram?

h=height

b=base

CUT HERE!

area of parallelogram2
Area of Parallelogram

What is the area

of this parallelogram?

h=height

b=base

MOVE TO HERE!

area of parallelogram3
Area of Parallelogram

It’s the same as

the area of this rectangle

h=height

b=base

area of parallelogram4
Area of Parallelogram

Arearectangle= base x height

(perpendicular height h)

h=height

b=base

area of parallelogram5
Area of Parallelogram

Areaparallelogram = base x height

(perpendicular height h)

h=height

b=base

area of parallelogram6
Area of Parallelogram
  • Words: Area =(base)(height)
  • Symbols: A = bh

h

b

Very Important: The height must be perpendicular to the base.

slide14

SOLUTION

Use the formula for the area of a parallelogram.

Substitute 9 for band 6 for h.

A = bh

Formula for the area of a parallelogram

=(9)(6)

Substitute 9 for b and 6 for h.

=54

Multiply.

The parallelogram has an area of 54

square meters.

ANSWER

Example 1

Find the Area of a Parallelogram

Find the area of the parallelogram.

slide15

A =bh

Formula for the area of a parallelogram

78=12h

Substitute 78 for A and 12 for b.

6.5 =h

Divide each side by 12.

The parallelogram has a height of 6.5 feet.

ANSWER

Example 2

Find the Height of a Parallelogram

Find the height of the parallelogram

given that its area is 78 square feet.

SOLUTION

slide16

Your Turn:

1.

96 yd2

ANSWER

2.

77 mm2

ANSWER

3.

196 ft2

ANSWER

Find the area of the parallelogram.

slide17

Your Turn:

4.

A = 72 in.2

h =6 in.

ANSWER

A = 30 m2

5.

b =6 m

ANSWER

6.

A = 28 cm2

h =4 cm

ANSWER

In Exercises 4–6, Agives the area of the parallelogram. Find the missing measure.

rhombus
Rhombus
  • A parallelogram with opposite equal acute and obtuse angles and four equal sides.

Diagonals

4 equal sides

Diagonals

area of a rhombus

h

b

Area of a Rhombus

Same as Area of parallelogram

A = b . h

special rhombus rule

d2

d1

Special “Rhombus Rule”
  • Since the diagonals are perpendicular
  • Another way to find the area of a rhombus is:
  • Area = ½ (product of the diagonals)

area = ½ (d1.d2)

This is good when you only know the diagonals, but not the sides or height

area of a rhombus1
Area of a Rhombus

The formula for the area of a rhombus can be justified using the area of a triangle. A specific case follows.

  • The diagonals divide a rhombus into 4 congruent right triangles. So, the area of the rhombus is 4 times the area of one of the right triangles.
  • Area of 1 triangle = 1/2bh = ½(3)(4) = 6
  • Area of 4 triangles = 4(6) = 24
  • Notice that 1/2d1d2 or ½(6)(8), also equals 24
slide22

Find the area of the rhombus.

b.

a.

1

1

1

1

1

2

2

2

2

2

Example 3

Find the Area of a Rhombus

SOLUTION

SOLUTION

a.

b.

A=d1d2

A=d1d2

=

(6 + 6)(9 + 9)

=

(14)(10)

= 70

=

(12)(18)

=108

The area of the rhombus is 70 square inches.

The area of the rhombus

is 108 square inches.